src/HOL/Complex/NSComplex.thy
author paulson
Thu Jul 01 12:29:53 2004 +0200 (2004-07-01)
changeset 15013 34264f5e4691
parent 15003 6145dd7538d7
child 15085 5693a977a767
permissions -rw-r--r--
new treatment of binary numerals
paulson@13957
     1
(*  Title:       NSComplex.thy
paulson@14430
     2
    ID:      $Id$
paulson@13957
     3
    Author:      Jacques D. Fleuriot
paulson@13957
     4
    Copyright:   2001  University of Edinburgh
paulson@14430
     5
    Conversion to Isar and new proofs by Lawrence C Paulson, 2003/4
paulson@13957
     6
*)
paulson@13957
     7
paulson@14430
     8
header{*Nonstandard Complex Numbers*}
paulson@14430
     9
paulson@14641
    10
theory NSComplex = Complex:
paulson@13957
    11
paulson@13957
    12
constdefs
paulson@13957
    13
    hcomplexrel :: "((nat=>complex)*(nat=>complex)) set"
paulson@14354
    14
    "hcomplexrel == {p. \<exists>X Y. p = ((X::nat=>complex),Y) &
paulson@13957
    15
                        {n::nat. X(n) = Y(n)}: FreeUltrafilterNat}"
paulson@13957
    16
paulson@14314
    17
typedef hcomplex = "{x::nat=>complex. True}//hcomplexrel"
paulson@14314
    18
  by (auto simp add: quotient_def)
paulson@13957
    19
wenzelm@14691
    20
instance hcomplex :: "{zero, one, plus, times, minus, inverse, power}" ..
paulson@14314
    21
paulson@14314
    22
defs (overloaded)
paulson@14314
    23
  hcomplex_zero_def:
paulson@13957
    24
  "0 == Abs_hcomplex(hcomplexrel `` {%n. (0::complex)})"
paulson@14314
    25
paulson@14314
    26
  hcomplex_one_def:
paulson@13957
    27
  "1 == Abs_hcomplex(hcomplexrel `` {%n. (1::complex)})"
paulson@13957
    28
paulson@13957
    29
paulson@14314
    30
  hcomplex_minus_def:
paulson@14314
    31
  "- z == Abs_hcomplex(UN X: Rep_hcomplex(z).
paulson@14314
    32
                       hcomplexrel `` {%n::nat. - (X n)})"
paulson@13957
    33
paulson@14314
    34
  hcomplex_diff_def:
paulson@13957
    35
  "w - z == w + -(z::hcomplex)"
paulson@14314
    36
paulson@14377
    37
  hcinv_def:
paulson@14377
    38
  "inverse(P) == Abs_hcomplex(UN X: Rep_hcomplex(P).
paulson@14377
    39
                    hcomplexrel `` {%n. inverse(X n)})"
paulson@14377
    40
paulson@13957
    41
constdefs
paulson@13957
    42
paulson@14314
    43
  hcomplex_of_complex :: "complex => hcomplex"
paulson@13957
    44
  "hcomplex_of_complex z == Abs_hcomplex(hcomplexrel `` {%n. z})"
paulson@14314
    45
paulson@13957
    46
  (*--- real and Imaginary parts ---*)
paulson@14314
    47
paulson@14314
    48
  hRe :: "hcomplex => hypreal"
paulson@13957
    49
  "hRe(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Re (X n)})"
paulson@13957
    50
paulson@14314
    51
  hIm :: "hcomplex => hypreal"
paulson@14314
    52
  "hIm(z) == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. Im (X n)})"
paulson@13957
    53
paulson@13957
    54
paulson@13957
    55
  (*----------- modulus ------------*)
paulson@13957
    56
paulson@14314
    57
  hcmod :: "hcomplex => hypreal"
paulson@13957
    58
  "hcmod z == Abs_hypreal(UN X: Rep_hcomplex(z).
paulson@13957
    59
			  hyprel `` {%n. cmod (X n)})"
paulson@13957
    60
paulson@14314
    61
  (*------ imaginary unit ----------*)
paulson@14314
    62
paulson@14314
    63
  iii :: hcomplex
paulson@13957
    64
  "iii == Abs_hcomplex(hcomplexrel `` {%n. ii})"
paulson@13957
    65
paulson@13957
    66
  (*------- complex conjugate ------*)
paulson@13957
    67
paulson@14314
    68
  hcnj :: "hcomplex => hcomplex"
paulson@13957
    69
  "hcnj z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. cnj (X n)})"
paulson@13957
    70
paulson@14314
    71
  (*------------ Argand -------------*)
paulson@13957
    72
paulson@14314
    73
  hsgn :: "hcomplex => hcomplex"
paulson@13957
    74
  "hsgn z == Abs_hcomplex(UN X:Rep_hcomplex(z). hcomplexrel `` {%n. sgn(X n)})"
paulson@13957
    75
paulson@14314
    76
  harg :: "hcomplex => hypreal"
paulson@13957
    77
  "harg z == Abs_hypreal(UN X:Rep_hcomplex(z). hyprel `` {%n. arg(X n)})"
paulson@13957
    78
paulson@13957
    79
  (* abbreviation for (cos a + i sin a) *)
paulson@14314
    80
  hcis :: "hypreal => hcomplex"
paulson@13957
    81
  "hcis a == Abs_hcomplex(UN X:Rep_hypreal(a). hcomplexrel `` {%n. cis (X n)})"
paulson@13957
    82
paulson@14314
    83
  (*----- injection from hyperreals -----*)
paulson@14314
    84
paulson@14314
    85
  hcomplex_of_hypreal :: "hypreal => hcomplex"
paulson@13957
    86
  "hcomplex_of_hypreal r == Abs_hcomplex(UN X:Rep_hypreal(r).
paulson@13957
    87
			       hcomplexrel `` {%n. complex_of_real (X n)})"
paulson@13957
    88
wenzelm@14653
    89
  (* abbreviation for r*(cos a + i sin a) *)
wenzelm@14653
    90
  hrcis :: "[hypreal, hypreal] => hcomplex"
wenzelm@14653
    91
  "hrcis r a == hcomplex_of_hypreal r * hcis a"
wenzelm@14653
    92
paulson@13957
    93
  (*------------ e ^ (x + iy) ------------*)
paulson@13957
    94
paulson@14314
    95
  hexpi :: "hcomplex => hcomplex"
paulson@13957
    96
  "hexpi z == hcomplex_of_hypreal(( *f* exp) (hRe z)) * hcis (hIm z)"
paulson@14314
    97
paulson@13957
    98
paulson@14377
    99
constdefs
paulson@14377
   100
  HComplex :: "[hypreal,hypreal] => hcomplex"
paulson@14377
   101
   "HComplex x y == hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y"
paulson@14377
   102
paulson@14377
   103
paulson@14314
   104
defs (overloaded)
paulson@13957
   105
paulson@13957
   106
  (*----------- division ----------*)
paulson@13957
   107
paulson@14314
   108
  hcomplex_divide_def:
paulson@13957
   109
  "w / (z::hcomplex) == w * inverse z"
paulson@14314
   110
paulson@14314
   111
  hcomplex_add_def:
paulson@13957
   112
  "w + z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
paulson@13957
   113
		      hcomplexrel `` {%n. X n + Y n})"
paulson@13957
   114
paulson@14314
   115
  hcomplex_mult_def:
paulson@13957
   116
  "w * z == Abs_hcomplex(UN X:Rep_hcomplex(w). UN Y:Rep_hcomplex(z).
paulson@14314
   117
		      hcomplexrel `` {%n. X n * Y n})"
paulson@13957
   118
paulson@13957
   119
paulson@13957
   120
paulson@13957
   121
consts
paulson@14314
   122
  "hcpow"  :: "[hcomplex,hypnat] => hcomplex"     (infixr 80)
paulson@13957
   123
paulson@13957
   124
defs
paulson@13957
   125
  (* hypernatural powers of nonstandard complex numbers *)
paulson@14314
   126
  hcpow_def:
paulson@14314
   127
  "(z::hcomplex) hcpow (n::hypnat)
paulson@13957
   128
      == Abs_hcomplex(UN X:Rep_hcomplex(z). UN Y: Rep_hypnat(n).
paulson@13957
   129
             hcomplexrel `` {%n. (X n) ^ (Y n)})"
paulson@13957
   130
paulson@14314
   131
paulson@14314
   132
lemma hcomplexrel_refl: "(x,x): hcomplexrel"
paulson@14374
   133
by (simp add: hcomplexrel_def)
paulson@14314
   134
paulson@14314
   135
lemma hcomplexrel_sym: "(x,y): hcomplexrel ==> (y,x):hcomplexrel"
paulson@14374
   136
by (auto simp add: hcomplexrel_def eq_commute)
paulson@14314
   137
paulson@14314
   138
lemma hcomplexrel_trans:
paulson@14314
   139
      "[|(x,y): hcomplexrel; (y,z):hcomplexrel|] ==> (x,z):hcomplexrel"
paulson@14374
   140
by (simp add: hcomplexrel_def, ultra)
paulson@14314
   141
paulson@14314
   142
lemma equiv_hcomplexrel: "equiv UNIV hcomplexrel"
paulson@14374
   143
apply (simp add: equiv_def refl_def sym_def trans_def hcomplexrel_refl)
paulson@14374
   144
apply (blast intro: hcomplexrel_sym hcomplexrel_trans)
paulson@14314
   145
done
paulson@14314
   146
paulson@14314
   147
lemmas equiv_hcomplexrel_iff =
paulson@14314
   148
    eq_equiv_class_iff [OF equiv_hcomplexrel UNIV_I UNIV_I, simp]
paulson@14314
   149
paulson@14314
   150
lemma hcomplexrel_in_hcomplex [simp]: "hcomplexrel``{x} : hcomplex"
paulson@14374
   151
by (simp add: hcomplex_def hcomplexrel_def quotient_def, blast)
paulson@14314
   152
paulson@14314
   153
lemma inj_on_Abs_hcomplex: "inj_on Abs_hcomplex hcomplex"
paulson@14314
   154
apply (rule inj_on_inverseI)
paulson@14314
   155
apply (erule Abs_hcomplex_inverse)
paulson@14314
   156
done
paulson@14314
   157
paulson@14314
   158
declare inj_on_Abs_hcomplex [THEN inj_on_iff, simp]
paulson@14314
   159
        Abs_hcomplex_inverse [simp]
paulson@14314
   160
paulson@14314
   161
declare equiv_hcomplexrel [THEN eq_equiv_class_iff, simp]
paulson@14314
   162
paulson@14314
   163
paulson@14314
   164
lemma inj_Rep_hcomplex: "inj(Rep_hcomplex)"
paulson@14314
   165
apply (rule inj_on_inverseI)
paulson@14314
   166
apply (rule Rep_hcomplex_inverse)
paulson@14314
   167
done
paulson@14314
   168
paulson@14374
   169
lemma lemma_hcomplexrel_refl [simp]: "x: hcomplexrel `` {x}"
paulson@14374
   170
by (simp add: hcomplexrel_def)
paulson@14314
   171
paulson@14374
   172
lemma hcomplex_empty_not_mem [simp]: "{} \<notin> hcomplex"
paulson@14374
   173
apply (simp add: hcomplex_def hcomplexrel_def)
paulson@14314
   174
apply (auto elim!: quotientE)
paulson@14314
   175
done
paulson@14314
   176
paulson@14374
   177
lemma Rep_hcomplex_nonempty [simp]: "Rep_hcomplex x \<noteq> {}"
paulson@14374
   178
by (cut_tac x = x in Rep_hcomplex, auto)
paulson@14314
   179
paulson@14314
   180
lemma eq_Abs_hcomplex:
paulson@14314
   181
    "(!!x. z = Abs_hcomplex(hcomplexrel `` {x}) ==> P) ==> P"
paulson@14314
   182
apply (rule_tac x1=z in Rep_hcomplex [unfolded hcomplex_def, THEN quotientE])
paulson@14314
   183
apply (drule_tac f = Abs_hcomplex in arg_cong)
paulson@14374
   184
apply (force simp add: Rep_hcomplex_inverse hcomplexrel_def)
paulson@14314
   185
done
paulson@14314
   186
paulson@14469
   187
theorem hcomplex_cases [case_names Abs_hcomplex, cases type: hcomplex]:
paulson@14469
   188
    "(!!x. z = Abs_hcomplex(hcomplexrel``{x}) ==> P) ==> P"
paulson@14469
   189
by (rule eq_Abs_hcomplex [of z], blast)
paulson@14469
   190
paulson@14377
   191
lemma hcomplexrel_iff [simp]:
paulson@14374
   192
   "((X,Y): hcomplexrel) = ({n. X n = Y n}: FreeUltrafilterNat)"
paulson@14374
   193
by (simp add: hcomplexrel_def)
paulson@14374
   194
paulson@14314
   195
paulson@14314
   196
subsection{*Properties of Nonstandard Real and Imaginary Parts*}
paulson@14314
   197
paulson@14314
   198
lemma hRe:
paulson@14314
   199
     "hRe(Abs_hcomplex (hcomplexrel `` {X})) =
paulson@14314
   200
      Abs_hypreal(hyprel `` {%n. Re(X n)})"
paulson@14374
   201
apply (simp add: hRe_def)
paulson@14374
   202
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14377
   203
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   204
done
paulson@14314
   205
paulson@14314
   206
lemma hIm:
paulson@14314
   207
     "hIm(Abs_hcomplex (hcomplexrel `` {X})) =
paulson@14314
   208
      Abs_hypreal(hyprel `` {%n. Im(X n)})"
paulson@14374
   209
apply (simp add: hIm_def)
paulson@14374
   210
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14377
   211
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   212
done
paulson@14314
   213
paulson@14335
   214
lemma hcomplex_hRe_hIm_cancel_iff:
paulson@14335
   215
     "(w=z) = (hRe(w) = hRe(z) & hIm(w) = hIm(z))"
paulson@14469
   216
apply (cases z, cases w)
paulson@14377
   217
apply (auto simp add: hRe hIm complex_Re_Im_cancel_iff iff: hcomplexrel_iff)
paulson@14314
   218
apply (ultra+)
paulson@14314
   219
done
paulson@14314
   220
paulson@14377
   221
lemma hcomplex_equality [intro?]: "hRe z = hRe w ==> hIm z = hIm w ==> z = w"
paulson@14377
   222
by (simp add: hcomplex_hRe_hIm_cancel_iff) 
paulson@14377
   223
paulson@14374
   224
lemma hcomplex_hRe_zero [simp]: "hRe 0 = 0"
paulson@14374
   225
by (simp add: hcomplex_zero_def hRe hypreal_zero_num)
paulson@14314
   226
paulson@14374
   227
lemma hcomplex_hIm_zero [simp]: "hIm 0 = 0"
paulson@14374
   228
by (simp add: hcomplex_zero_def hIm hypreal_zero_num)
paulson@14314
   229
paulson@14374
   230
lemma hcomplex_hRe_one [simp]: "hRe 1 = 1"
paulson@14374
   231
by (simp add: hcomplex_one_def hRe hypreal_one_num)
paulson@14314
   232
paulson@14374
   233
lemma hcomplex_hIm_one [simp]: "hIm 1 = 0"
paulson@14374
   234
by (simp add: hcomplex_one_def hIm hypreal_one_def hypreal_zero_num)
paulson@14314
   235
paulson@14314
   236
paulson@14354
   237
subsection{*Addition for Nonstandard Complex Numbers*}
paulson@14314
   238
paulson@14314
   239
lemma hcomplex_add_congruent2:
paulson@14658
   240
    "congruent2 hcomplexrel hcomplexrel (%X Y. hcomplexrel `` {%n. X n + Y n})"
paulson@14377
   241
by (auto simp add: congruent2_def iff: hcomplexrel_iff, ultra) 
paulson@14314
   242
paulson@14314
   243
lemma hcomplex_add:
paulson@14377
   244
  "Abs_hcomplex(hcomplexrel``{%n. X n}) + 
paulson@14377
   245
   Abs_hcomplex(hcomplexrel``{%n. Y n}) =
paulson@14377
   246
     Abs_hcomplex(hcomplexrel``{%n. X n + Y n})"
paulson@14374
   247
apply (simp add: hcomplex_add_def)
paulson@14374
   248
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   249
apply (auto simp add: iff: hcomplexrel_iff, ultra) 
paulson@14314
   250
done
paulson@14314
   251
paulson@14314
   252
lemma hcomplex_add_commute: "(z::hcomplex) + w = w + z"
paulson@14469
   253
apply (cases z, cases w)
paulson@14335
   254
apply (simp add: complex_add_commute hcomplex_add)
paulson@14314
   255
done
paulson@14314
   256
paulson@14314
   257
lemma hcomplex_add_assoc: "((z1::hcomplex) + z2) + z3 = z1 + (z2 + z3)"
paulson@14469
   258
apply (cases z1, cases z2, cases z3)
paulson@14335
   259
apply (simp add: hcomplex_add complex_add_assoc)
paulson@14314
   260
done
paulson@14314
   261
paulson@14314
   262
lemma hcomplex_add_zero_left: "(0::hcomplex) + z = z"
paulson@14469
   263
apply (cases z)
paulson@14374
   264
apply (simp add: hcomplex_zero_def hcomplex_add)
paulson@14314
   265
done
paulson@14314
   266
paulson@14314
   267
lemma hcomplex_add_zero_right: "z + (0::hcomplex) = z"
paulson@14374
   268
by (simp add: hcomplex_add_zero_left hcomplex_add_commute)
paulson@14314
   269
paulson@14314
   270
lemma hRe_add: "hRe(x + y) = hRe(x) + hRe(y)"
paulson@14469
   271
apply (cases x, cases y)
paulson@14374
   272
apply (simp add: hRe hcomplex_add hypreal_add complex_Re_add)
paulson@14314
   273
done
paulson@14314
   274
paulson@14314
   275
lemma hIm_add: "hIm(x + y) = hIm(x) + hIm(y)"
paulson@14469
   276
apply (cases x, cases y)
paulson@14374
   277
apply (simp add: hIm hcomplex_add hypreal_add complex_Im_add)
paulson@14314
   278
done
paulson@14314
   279
paulson@14354
   280
paulson@14354
   281
subsection{*Additive Inverse on Nonstandard Complex Numbers*}
paulson@14314
   282
paulson@14314
   283
lemma hcomplex_minus_congruent:
paulson@14374
   284
     "congruent hcomplexrel (%X. hcomplexrel `` {%n. - (X n)})"
paulson@14374
   285
by (simp add: congruent_def)
paulson@14314
   286
paulson@14314
   287
lemma hcomplex_minus:
paulson@14314
   288
  "- (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
   289
      Abs_hcomplex(hcomplexrel `` {%n. -(X n)})"
paulson@14374
   290
apply (simp add: hcomplex_minus_def)
paulson@14374
   291
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   292
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   293
done
paulson@14314
   294
paulson@14314
   295
lemma hcomplex_add_minus_left: "-z + z = (0::hcomplex)"
paulson@14469
   296
apply (cases z)
paulson@14374
   297
apply (simp add: hcomplex_add hcomplex_minus hcomplex_zero_def)
paulson@14314
   298
done
paulson@14335
   299
paulson@14314
   300
paulson@14314
   301
subsection{*Multiplication for Nonstandard Complex Numbers*}
paulson@14314
   302
paulson@14314
   303
lemma hcomplex_mult:
paulson@14374
   304
  "Abs_hcomplex(hcomplexrel``{%n. X n}) *
paulson@14335
   305
     Abs_hcomplex(hcomplexrel``{%n. Y n}) =
paulson@14374
   306
     Abs_hcomplex(hcomplexrel``{%n. X n * Y n})"
paulson@14374
   307
apply (simp add: hcomplex_mult_def)
paulson@14374
   308
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   309
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   310
done
paulson@14314
   311
paulson@14314
   312
lemma hcomplex_mult_commute: "(w::hcomplex) * z = z * w"
paulson@14469
   313
apply (cases w, cases z)
paulson@14374
   314
apply (simp add: hcomplex_mult complex_mult_commute)
paulson@14314
   315
done
paulson@14314
   316
paulson@14314
   317
lemma hcomplex_mult_assoc: "((u::hcomplex) * v) * w = u * (v * w)"
paulson@14469
   318
apply (cases u, cases v, cases w)
paulson@14374
   319
apply (simp add: hcomplex_mult complex_mult_assoc)
paulson@14314
   320
done
paulson@14314
   321
paulson@14314
   322
lemma hcomplex_mult_one_left: "(1::hcomplex) * z = z"
paulson@14469
   323
apply (cases z)
paulson@14374
   324
apply (simp add: hcomplex_one_def hcomplex_mult)
paulson@14314
   325
done
paulson@14314
   326
paulson@14314
   327
lemma hcomplex_mult_zero_left: "(0::hcomplex) * z = 0"
paulson@14469
   328
apply (cases z)
paulson@14374
   329
apply (simp add: hcomplex_zero_def hcomplex_mult)
paulson@14314
   330
done
paulson@14314
   331
paulson@14335
   332
lemma hcomplex_add_mult_distrib:
paulson@14335
   333
     "((z1::hcomplex) + z2) * w = (z1 * w) + (z2 * w)"
paulson@14469
   334
apply (cases z1, cases z2, cases w)
paulson@14374
   335
apply (simp add: hcomplex_mult hcomplex_add left_distrib)
paulson@14314
   336
done
paulson@14314
   337
paulson@14354
   338
lemma hcomplex_zero_not_eq_one: "(0::hcomplex) \<noteq> (1::hcomplex)"
paulson@14374
   339
by (simp add: hcomplex_zero_def hcomplex_one_def)
paulson@14374
   340
paulson@14314
   341
declare hcomplex_zero_not_eq_one [THEN not_sym, simp]
paulson@14314
   342
paulson@14314
   343
paulson@14314
   344
subsection{*Inverse of Nonstandard Complex Number*}
paulson@14314
   345
paulson@14314
   346
lemma hcomplex_inverse:
paulson@14314
   347
  "inverse (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
   348
      Abs_hcomplex(hcomplexrel `` {%n. inverse (X n)})"
paulson@14374
   349
apply (simp add: hcinv_def)
paulson@14374
   350
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   351
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   352
done
paulson@14314
   353
paulson@14314
   354
lemma hcomplex_mult_inv_left:
paulson@14354
   355
      "z \<noteq> (0::hcomplex) ==> inverse(z) * z = (1::hcomplex)"
paulson@14469
   356
apply (cases z)
paulson@14374
   357
apply (simp add: hcomplex_zero_def hcomplex_one_def hcomplex_inverse hcomplex_mult, ultra)
paulson@14314
   358
apply (rule ccontr)
paulson@14374
   359
apply (drule left_inverse, auto)
paulson@14314
   360
done
paulson@14314
   361
paulson@14318
   362
subsection {* The Field of Nonstandard Complex Numbers *}
paulson@14318
   363
paulson@14318
   364
instance hcomplex :: field
paulson@14318
   365
proof
paulson@14318
   366
  fix z u v w :: hcomplex
paulson@14318
   367
  show "(u + v) + w = u + (v + w)"
paulson@14318
   368
    by (simp add: hcomplex_add_assoc)
paulson@14318
   369
  show "z + w = w + z"
paulson@14318
   370
    by (simp add: hcomplex_add_commute)
paulson@14318
   371
  show "0 + z = z"
paulson@14335
   372
    by (simp add: hcomplex_add_zero_left)
paulson@14318
   373
  show "-z + z = 0"
paulson@14335
   374
    by (simp add: hcomplex_add_minus_left)
paulson@14318
   375
  show "z - w = z + -w"
paulson@14318
   376
    by (simp add: hcomplex_diff_def)
paulson@14318
   377
  show "(u * v) * w = u * (v * w)"
paulson@14318
   378
    by (simp add: hcomplex_mult_assoc)
paulson@14318
   379
  show "z * w = w * z"
paulson@14318
   380
    by (simp add: hcomplex_mult_commute)
paulson@14318
   381
  show "1 * z = z"
paulson@14335
   382
    by (simp add: hcomplex_mult_one_left)
paulson@14318
   383
  show "0 \<noteq> (1::hcomplex)"
paulson@14318
   384
    by (rule hcomplex_zero_not_eq_one)
paulson@14318
   385
  show "(u + v) * w = u * w + v * w"
paulson@14318
   386
    by (simp add: hcomplex_add_mult_distrib)
paulson@14430
   387
  show "z / w = z * inverse w"
paulson@14318
   388
    by (simp add: hcomplex_divide_def)
paulson@14430
   389
  assume "w \<noteq> 0"
paulson@14430
   390
  thus "inverse w * w = 1"
paulson@14318
   391
    by (rule hcomplex_mult_inv_left)
paulson@14318
   392
qed
paulson@14318
   393
paulson@14318
   394
instance hcomplex :: division_by_zero
paulson@14318
   395
proof
paulson@14430
   396
  show "inverse 0 = (0::hcomplex)"
paulson@14374
   397
    by (simp add: hcomplex_inverse hcomplex_zero_def)
paulson@14318
   398
qed
paulson@14314
   399
paulson@14374
   400
paulson@14318
   401
subsection{*More Minus Laws*}
paulson@14318
   402
paulson@14318
   403
lemma hRe_minus: "hRe(-z) = - hRe(z)"
paulson@14469
   404
apply (cases z)
paulson@14374
   405
apply (simp add: hRe hcomplex_minus hypreal_minus complex_Re_minus)
paulson@14318
   406
done
paulson@14318
   407
paulson@14318
   408
lemma hIm_minus: "hIm(-z) = - hIm(z)"
paulson@14469
   409
apply (cases z)
paulson@14374
   410
apply (simp add: hIm hcomplex_minus hypreal_minus complex_Im_minus)
paulson@14318
   411
done
paulson@14318
   412
paulson@14318
   413
lemma hcomplex_add_minus_eq_minus:
paulson@14318
   414
      "x + y = (0::hcomplex) ==> x = -y"
obua@14738
   415
apply (drule OrderedGroup.equals_zero_I)
paulson@14374
   416
apply (simp add: minus_equation_iff [of x y])
paulson@14318
   417
done
paulson@14318
   418
paulson@14377
   419
lemma hcomplex_i_mult_eq [simp]: "iii * iii = - 1"
paulson@14377
   420
by (simp add: iii_def hcomplex_mult hcomplex_one_def hcomplex_minus)
paulson@14377
   421
paulson@14377
   422
lemma hcomplex_i_mult_left [simp]: "iii * (iii * z) = -z"
paulson@14377
   423
by (simp add: mult_assoc [symmetric])
paulson@14377
   424
paulson@14377
   425
lemma hcomplex_i_not_zero [simp]: "iii \<noteq> 0"
paulson@14377
   426
by (simp add: iii_def hcomplex_zero_def)
paulson@14377
   427
paulson@14318
   428
paulson@14318
   429
subsection{*More Multiplication Laws*}
paulson@14318
   430
paulson@14318
   431
lemma hcomplex_mult_one_right: "z * (1::hcomplex) = z"
obua@14738
   432
by (rule OrderedGroup.mult_1_right)
paulson@14318
   433
paulson@14374
   434
lemma hcomplex_mult_minus_one [simp]: "- 1 * (z::hcomplex) = -z"
paulson@14374
   435
by simp
paulson@14318
   436
paulson@14374
   437
lemma hcomplex_mult_minus_one_right [simp]: "(z::hcomplex) * - 1 = -z"
paulson@14374
   438
by (subst hcomplex_mult_commute, simp)
paulson@14318
   439
paulson@14335
   440
lemma hcomplex_mult_left_cancel:
paulson@14354
   441
     "(c::hcomplex) \<noteq> (0::hcomplex) ==> (c*a=c*b) = (a=b)"
paulson@14374
   442
by (simp add: field_mult_cancel_left)
paulson@14314
   443
paulson@14335
   444
lemma hcomplex_mult_right_cancel:
paulson@14354
   445
     "(c::hcomplex) \<noteq> (0::hcomplex) ==> (a*c=b*c) = (a=b)"
paulson@14374
   446
by (simp add: Ring_and_Field.field_mult_cancel_right)
paulson@14314
   447
paulson@14314
   448
paulson@14318
   449
subsection{*Subraction and Division*}
paulson@14314
   450
paulson@14318
   451
lemma hcomplex_diff:
paulson@14318
   452
 "Abs_hcomplex(hcomplexrel``{%n. X n}) - Abs_hcomplex(hcomplexrel``{%n. Y n}) =
paulson@14318
   453
  Abs_hcomplex(hcomplexrel``{%n. X n - Y n})"
paulson@14374
   454
by (simp add: hcomplex_diff_def hcomplex_minus hcomplex_add complex_diff_def)
paulson@14314
   455
paulson@14374
   456
lemma hcomplex_diff_eq_eq [simp]: "((x::hcomplex) - y = z) = (x = z + y)"
obua@14738
   457
by (rule OrderedGroup.diff_eq_eq)
paulson@14314
   458
paulson@14314
   459
lemma hcomplex_add_divide_distrib: "(x+y)/(z::hcomplex) = x/z + y/z"
paulson@14374
   460
by (rule Ring_and_Field.add_divide_distrib)
paulson@14314
   461
paulson@14314
   462
paulson@14314
   463
subsection{*Embedding Properties for @{term hcomplex_of_hypreal} Map*}
paulson@14314
   464
paulson@14314
   465
lemma hcomplex_of_hypreal:
paulson@14314
   466
  "hcomplex_of_hypreal (Abs_hypreal(hyprel `` {%n. X n})) =
paulson@14314
   467
      Abs_hcomplex(hcomplexrel `` {%n. complex_of_real (X n)})"
paulson@14374
   468
apply (simp add: hcomplex_of_hypreal_def)
paulson@14377
   469
apply (rule_tac f = Abs_hcomplex in arg_cong, auto iff: hcomplexrel_iff, ultra)
paulson@14314
   470
done
paulson@14314
   471
paulson@14374
   472
lemma hcomplex_of_hypreal_cancel_iff [iff]:
paulson@14374
   473
     "(hcomplex_of_hypreal x = hcomplex_of_hypreal y) = (x = y)"
paulson@14469
   474
apply (cases x, cases y)
paulson@14374
   475
apply (simp add: hcomplex_of_hypreal)
paulson@14314
   476
done
paulson@14314
   477
paulson@14374
   478
lemma hcomplex_of_hypreal_one [simp]: "hcomplex_of_hypreal 1 = 1"
paulson@14374
   479
by (simp add: hcomplex_one_def hcomplex_of_hypreal hypreal_one_num)
paulson@14314
   480
paulson@14374
   481
lemma hcomplex_of_hypreal_zero [simp]: "hcomplex_of_hypreal 0 = 0"
paulson@14374
   482
by (simp add: hcomplex_zero_def hypreal_zero_def hcomplex_of_hypreal)
paulson@14374
   483
paulson@15013
   484
lemma hcomplex_of_hypreal_minus [simp]:
paulson@15013
   485
     "hcomplex_of_hypreal(-x) = - hcomplex_of_hypreal x"
paulson@15013
   486
apply (cases x)
paulson@15013
   487
apply (simp add: hcomplex_of_hypreal hcomplex_minus hypreal_minus)
paulson@15013
   488
done
paulson@15013
   489
paulson@15013
   490
lemma hcomplex_of_hypreal_inverse [simp]:
paulson@15013
   491
     "hcomplex_of_hypreal(inverse x) = inverse(hcomplex_of_hypreal x)"
paulson@15013
   492
apply (cases x)
paulson@15013
   493
apply (simp add: hcomplex_of_hypreal hypreal_inverse hcomplex_inverse)
paulson@15013
   494
done
paulson@15013
   495
paulson@15013
   496
lemma hcomplex_of_hypreal_add [simp]:
paulson@15013
   497
  "hcomplex_of_hypreal (x + y) = hcomplex_of_hypreal x + hcomplex_of_hypreal y"
paulson@15013
   498
apply (cases x, cases y)
paulson@15013
   499
apply (simp add: hcomplex_of_hypreal hypreal_add hcomplex_add)
paulson@15013
   500
done
paulson@15013
   501
paulson@15013
   502
lemma hcomplex_of_hypreal_diff [simp]:
paulson@15013
   503
     "hcomplex_of_hypreal (x - y) =
paulson@15013
   504
      hcomplex_of_hypreal x - hcomplex_of_hypreal y "
paulson@15013
   505
by (simp add: hcomplex_diff_def hypreal_diff_def)
paulson@15013
   506
paulson@15013
   507
lemma hcomplex_of_hypreal_mult [simp]:
paulson@15013
   508
  "hcomplex_of_hypreal (x * y) = hcomplex_of_hypreal x * hcomplex_of_hypreal y"
paulson@15013
   509
apply (cases x, cases y)
paulson@15013
   510
apply (simp add: hcomplex_of_hypreal hypreal_mult hcomplex_mult)
paulson@15013
   511
done
paulson@15013
   512
paulson@15013
   513
lemma hcomplex_of_hypreal_divide [simp]:
paulson@15013
   514
  "hcomplex_of_hypreal(x/y) = hcomplex_of_hypreal x / hcomplex_of_hypreal y"
paulson@15013
   515
apply (simp add: hcomplex_divide_def)
paulson@15013
   516
apply (case_tac "y=0", simp)
paulson@15013
   517
apply (simp add: hypreal_divide_def)
paulson@15013
   518
done
paulson@15013
   519
paulson@14374
   520
lemma hRe_hcomplex_of_hypreal [simp]: "hRe(hcomplex_of_hypreal z) = z"
paulson@14469
   521
apply (cases z)
paulson@14314
   522
apply (auto simp add: hcomplex_of_hypreal hRe)
paulson@14314
   523
done
paulson@14314
   524
paulson@14374
   525
lemma hIm_hcomplex_of_hypreal [simp]: "hIm(hcomplex_of_hypreal z) = 0"
paulson@14469
   526
apply (cases z)
paulson@14314
   527
apply (auto simp add: hcomplex_of_hypreal hIm hypreal_zero_num)
paulson@14314
   528
done
paulson@14314
   529
paulson@14374
   530
lemma hcomplex_of_hypreal_epsilon_not_zero [simp]:
paulson@14374
   531
     "hcomplex_of_hypreal epsilon \<noteq> 0"
paulson@14374
   532
by (auto simp add: hcomplex_of_hypreal epsilon_def hcomplex_zero_def)
paulson@14314
   533
paulson@14318
   534
paulson@14377
   535
subsection{*HComplex theorems*}
paulson@14377
   536
paulson@14377
   537
lemma hRe_HComplex [simp]: "hRe (HComplex x y) = x"
paulson@14469
   538
apply (cases x, cases y)
paulson@14377
   539
apply (simp add: HComplex_def hRe iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal)
paulson@14377
   540
done
paulson@14377
   541
paulson@14377
   542
lemma hIm_HComplex [simp]: "hIm (HComplex x y) = y"
paulson@14469
   543
apply (cases x, cases y)
paulson@14377
   544
apply (simp add: HComplex_def hIm iii_def hcomplex_add hcomplex_mult hcomplex_of_hypreal)
paulson@14377
   545
done
paulson@14377
   546
paulson@14377
   547
text{*Relates the two nonstandard constructions*}
paulson@14377
   548
lemma HComplex_eq_Abs_hcomplex_Complex:
paulson@14377
   549
     "HComplex (Abs_hypreal (hyprel `` {X})) (Abs_hypreal (hyprel `` {Y})) =
paulson@14377
   550
      Abs_hcomplex(hcomplexrel `` {%n::nat. Complex (X n) (Y n)})";
paulson@14377
   551
by (simp add: hcomplex_hRe_hIm_cancel_iff hRe hIm) 
paulson@14377
   552
paulson@14377
   553
lemma hcomplex_surj [simp]: "HComplex (hRe z) (hIm z) = z"
paulson@14377
   554
by (simp add: hcomplex_equality) 
paulson@14377
   555
paulson@14377
   556
lemma hcomplex_induct [case_names rect, induct type: hcomplex]:
paulson@14377
   557
     "(\<And>x y. P (HComplex x y)) ==> P z"
paulson@14377
   558
by (rule hcomplex_surj [THEN subst], blast)
paulson@14377
   559
paulson@14377
   560
paulson@14318
   561
subsection{*Modulus (Absolute Value) of Nonstandard Complex Number*}
paulson@14314
   562
paulson@14314
   563
lemma hcmod:
paulson@14314
   564
  "hcmod (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
   565
      Abs_hypreal(hyprel `` {%n. cmod (X n)})"
paulson@14314
   566
paulson@14374
   567
apply (simp add: hcmod_def)
paulson@14374
   568
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14377
   569
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   570
done
paulson@14314
   571
paulson@14374
   572
lemma hcmod_zero [simp]: "hcmod(0) = 0"
paulson@14377
   573
by (simp add: hcomplex_zero_def hypreal_zero_def hcmod)
paulson@14314
   574
paulson@14374
   575
lemma hcmod_one [simp]: "hcmod(1) = 1"
paulson@14374
   576
by (simp add: hcomplex_one_def hcmod hypreal_one_num)
paulson@14314
   577
paulson@14374
   578
lemma hcmod_hcomplex_of_hypreal [simp]: "hcmod(hcomplex_of_hypreal x) = abs x"
paulson@14469
   579
apply (cases x)
paulson@14314
   580
apply (auto simp add: hcmod hcomplex_of_hypreal hypreal_hrabs)
paulson@14314
   581
done
paulson@14314
   582
paulson@14335
   583
lemma hcomplex_of_hypreal_abs:
paulson@14335
   584
     "hcomplex_of_hypreal (abs x) =
paulson@14314
   585
      hcomplex_of_hypreal(hcmod(hcomplex_of_hypreal x))"
paulson@14374
   586
by simp
paulson@14314
   587
paulson@14377
   588
lemma HComplex_inject [simp]: "HComplex x y = HComplex x' y' = (x=x' & y=y')"
paulson@14377
   589
apply (rule iffI) 
paulson@14377
   590
 prefer 2 apply simp 
paulson@14377
   591
apply (simp add: HComplex_def iii_def) 
paulson@14469
   592
apply (cases x, cases y, cases x', cases y')
paulson@14377
   593
apply (auto simp add: iii_def hcomplex_mult hcomplex_add hcomplex_of_hypreal)
paulson@14377
   594
apply (ultra+) 
paulson@14377
   595
done
paulson@14377
   596
paulson@14377
   597
lemma HComplex_add [simp]:
paulson@14377
   598
     "HComplex x1 y1 + HComplex x2 y2 = HComplex (x1+x2) (y1+y2)"
paulson@15013
   599
by (simp add: HComplex_def add_ac right_distrib) 
paulson@14377
   600
paulson@14377
   601
lemma HComplex_minus [simp]: "- HComplex x y = HComplex (-x) (-y)"
paulson@14377
   602
by (simp add: HComplex_def hcomplex_of_hypreal_minus) 
paulson@14377
   603
paulson@14377
   604
lemma HComplex_diff [simp]:
paulson@14377
   605
     "HComplex x1 y1 - HComplex x2 y2 = HComplex (x1-x2) (y1-y2)"
paulson@14377
   606
by (simp add: diff_minus)
paulson@14377
   607
paulson@14377
   608
lemma HComplex_mult [simp]:
paulson@14377
   609
  "HComplex x1 y1 * HComplex x2 y2 = HComplex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
paulson@14377
   610
by (simp add: HComplex_def diff_minus hcomplex_of_hypreal_minus 
paulson@14377
   611
       add_ac mult_ac right_distrib)
paulson@14377
   612
paulson@14377
   613
(*HComplex_inverse is proved below*)
paulson@14377
   614
paulson@14377
   615
lemma hcomplex_of_hypreal_eq: "hcomplex_of_hypreal r = HComplex r 0"
paulson@14377
   616
by (simp add: HComplex_def)
paulson@14377
   617
paulson@14377
   618
lemma HComplex_add_hcomplex_of_hypreal [simp]:
paulson@14377
   619
     "HComplex x y + hcomplex_of_hypreal r = HComplex (x+r) y"
paulson@14377
   620
by (simp add: hcomplex_of_hypreal_eq)
paulson@14377
   621
paulson@14377
   622
lemma hcomplex_of_hypreal_add_HComplex [simp]:
paulson@14377
   623
     "hcomplex_of_hypreal r + HComplex x y = HComplex (r+x) y"
paulson@14377
   624
by (simp add: i_def hcomplex_of_hypreal_eq)
paulson@14377
   625
paulson@14377
   626
lemma HComplex_mult_hcomplex_of_hypreal:
paulson@14377
   627
     "HComplex x y * hcomplex_of_hypreal r = HComplex (x*r) (y*r)"
paulson@14377
   628
by (simp add: hcomplex_of_hypreal_eq)
paulson@14377
   629
paulson@14377
   630
lemma hcomplex_of_hypreal_mult_HComplex:
paulson@14377
   631
     "hcomplex_of_hypreal r * HComplex x y = HComplex (r*x) (r*y)"
paulson@14377
   632
by (simp add: i_def hcomplex_of_hypreal_eq)
paulson@14377
   633
paulson@14377
   634
lemma i_hcomplex_of_hypreal [simp]:
paulson@14377
   635
     "iii * hcomplex_of_hypreal r = HComplex 0 r"
paulson@14377
   636
by (simp add: HComplex_def)
paulson@14377
   637
paulson@14377
   638
lemma hcomplex_of_hypreal_i [simp]:
paulson@14377
   639
     "hcomplex_of_hypreal r * iii = HComplex 0 r"
paulson@14377
   640
by (simp add: mult_commute) 
paulson@14377
   641
paulson@14314
   642
paulson@14314
   643
subsection{*Conjugation*}
paulson@14314
   644
paulson@14314
   645
lemma hcnj:
paulson@14314
   646
  "hcnj (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14318
   647
   Abs_hcomplex(hcomplexrel `` {%n. cnj(X n)})"
paulson@14374
   648
apply (simp add: hcnj_def)
paulson@14374
   649
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   650
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   651
done
paulson@14314
   652
paulson@14374
   653
lemma hcomplex_hcnj_cancel_iff [iff]: "(hcnj x = hcnj y) = (x = y)"
paulson@14469
   654
apply (cases x, cases y)
paulson@14374
   655
apply (simp add: hcnj)
paulson@14374
   656
done
paulson@14374
   657
paulson@14374
   658
lemma hcomplex_hcnj_hcnj [simp]: "hcnj (hcnj z) = z"
paulson@14469
   659
apply (cases z)
paulson@14374
   660
apply (simp add: hcnj)
paulson@14314
   661
done
paulson@14314
   662
paulson@14374
   663
lemma hcomplex_hcnj_hcomplex_of_hypreal [simp]:
paulson@14374
   664
     "hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
paulson@14469
   665
apply (cases x)
paulson@14374
   666
apply (simp add: hcnj hcomplex_of_hypreal)
paulson@14314
   667
done
paulson@14314
   668
paulson@14374
   669
lemma hcomplex_hmod_hcnj [simp]: "hcmod (hcnj z) = hcmod z"
paulson@14469
   670
apply (cases z)
paulson@14374
   671
apply (simp add: hcnj hcmod)
paulson@14314
   672
done
paulson@14314
   673
paulson@14314
   674
lemma hcomplex_hcnj_minus: "hcnj (-z) = - hcnj z"
paulson@14469
   675
apply (cases z)
paulson@14374
   676
apply (simp add: hcnj hcomplex_minus complex_cnj_minus)
paulson@14314
   677
done
paulson@14314
   678
paulson@14314
   679
lemma hcomplex_hcnj_inverse: "hcnj(inverse z) = inverse(hcnj z)"
paulson@14469
   680
apply (cases z)
paulson@14374
   681
apply (simp add: hcnj hcomplex_inverse complex_cnj_inverse)
paulson@14314
   682
done
paulson@14314
   683
paulson@14314
   684
lemma hcomplex_hcnj_add: "hcnj(w + z) = hcnj(w) + hcnj(z)"
paulson@14469
   685
apply (cases z, cases w)
paulson@14374
   686
apply (simp add: hcnj hcomplex_add complex_cnj_add)
paulson@14314
   687
done
paulson@14314
   688
paulson@14314
   689
lemma hcomplex_hcnj_diff: "hcnj(w - z) = hcnj(w) - hcnj(z)"
paulson@14469
   690
apply (cases z, cases w)
paulson@14374
   691
apply (simp add: hcnj hcomplex_diff complex_cnj_diff)
paulson@14314
   692
done
paulson@14314
   693
paulson@14314
   694
lemma hcomplex_hcnj_mult: "hcnj(w * z) = hcnj(w) * hcnj(z)"
paulson@14469
   695
apply (cases z, cases w)
paulson@14374
   696
apply (simp add: hcnj hcomplex_mult complex_cnj_mult)
paulson@14314
   697
done
paulson@14314
   698
paulson@14314
   699
lemma hcomplex_hcnj_divide: "hcnj(w / z) = (hcnj w)/(hcnj z)"
paulson@14374
   700
by (simp add: hcomplex_divide_def hcomplex_hcnj_mult hcomplex_hcnj_inverse)
paulson@14314
   701
paulson@14374
   702
lemma hcnj_one [simp]: "hcnj 1 = 1"
paulson@14374
   703
by (simp add: hcomplex_one_def hcnj)
paulson@14314
   704
paulson@14374
   705
lemma hcomplex_hcnj_zero [simp]: "hcnj 0 = 0"
paulson@14374
   706
by (simp add: hcomplex_zero_def hcnj)
paulson@14374
   707
paulson@14374
   708
lemma hcomplex_hcnj_zero_iff [iff]: "(hcnj z = 0) = (z = 0)"
paulson@14469
   709
apply (cases z)
paulson@14374
   710
apply (simp add: hcomplex_zero_def hcnj)
paulson@14314
   711
done
paulson@14314
   712
paulson@14335
   713
lemma hcomplex_mult_hcnj:
paulson@14335
   714
     "z * hcnj z = hcomplex_of_hypreal (hRe(z) ^ 2 + hIm(z) ^ 2)"
paulson@14469
   715
apply (cases z)
paulson@14374
   716
apply (simp add: hcnj hcomplex_mult hcomplex_of_hypreal hRe hIm hypreal_add
paulson@14374
   717
                      hypreal_mult complex_mult_cnj numeral_2_eq_2)
paulson@14314
   718
done
paulson@14314
   719
paulson@14314
   720
paulson@14354
   721
subsection{*More Theorems about the Function @{term hcmod}*}
paulson@14314
   722
paulson@14374
   723
lemma hcomplex_hcmod_eq_zero_cancel [simp]: "(hcmod x = 0) = (x = 0)"
paulson@14469
   724
apply (cases x)
paulson@14374
   725
apply (simp add: hcmod hcomplex_zero_def hypreal_zero_num)
paulson@14314
   726
done
paulson@14314
   727
paulson@14374
   728
lemma hcmod_hcomplex_of_hypreal_of_nat [simp]:
paulson@14335
   729
     "hcmod (hcomplex_of_hypreal(hypreal_of_nat n)) = hypreal_of_nat n"
paulson@14374
   730
apply (simp add: abs_if linorder_not_less)
paulson@14314
   731
done
paulson@14314
   732
paulson@14374
   733
lemma hcmod_hcomplex_of_hypreal_of_hypnat [simp]:
paulson@14335
   734
     "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n"
paulson@14374
   735
apply (simp add: abs_if linorder_not_less)
paulson@14314
   736
done
paulson@14314
   737
paulson@14374
   738
lemma hcmod_minus [simp]: "hcmod (-x) = hcmod(x)"
paulson@14469
   739
apply (cases x)
paulson@14374
   740
apply (simp add: hcmod hcomplex_minus)
paulson@14314
   741
done
paulson@14314
   742
paulson@14314
   743
lemma hcmod_mult_hcnj: "hcmod(z * hcnj(z)) = hcmod(z) ^ 2"
paulson@14469
   744
apply (cases z)
paulson@14374
   745
apply (simp add: hcmod hcomplex_mult hcnj hypreal_mult complex_mod_mult_cnj numeral_2_eq_2)
paulson@14314
   746
done
paulson@14314
   747
paulson@14374
   748
lemma hcmod_ge_zero [simp]: "(0::hypreal) \<le> hcmod x"
paulson@14469
   749
apply (cases x)
paulson@14374
   750
apply (simp add: hcmod hypreal_zero_num hypreal_le)
paulson@14314
   751
done
paulson@14314
   752
paulson@14374
   753
lemma hrabs_hcmod_cancel [simp]: "abs(hcmod x) = hcmod x"
paulson@14374
   754
by (simp add: abs_if linorder_not_less)
paulson@14314
   755
paulson@14314
   756
lemma hcmod_mult: "hcmod(x*y) = hcmod(x) * hcmod(y)"
paulson@14469
   757
apply (cases x, cases y)
paulson@14374
   758
apply (simp add: hcmod hcomplex_mult hypreal_mult complex_mod_mult)
paulson@14314
   759
done
paulson@14314
   760
paulson@14314
   761
lemma hcmod_add_squared_eq:
paulson@14314
   762
     "hcmod(x + y) ^ 2 = hcmod(x) ^ 2 + hcmod(y) ^ 2 + 2 * hRe(x * hcnj y)"
paulson@14469
   763
apply (cases x, cases y)
paulson@14374
   764
apply (simp add: hcmod hcomplex_add hypreal_mult hRe hcnj hcomplex_mult
paulson@14374
   765
                      numeral_2_eq_2 realpow_two [symmetric]
paulson@14374
   766
                  del: realpow_Suc)
paulson@14374
   767
apply (simp add: numeral_2_eq_2 [symmetric] complex_mod_add_squared_eq
paulson@14374
   768
                 hypreal_add [symmetric] hypreal_mult [symmetric]
paulson@14314
   769
                 hypreal_of_real_def [symmetric])
paulson@14314
   770
done
paulson@14314
   771
paulson@14374
   772
lemma hcomplex_hRe_mult_hcnj_le_hcmod [simp]: "hRe(x * hcnj y) \<le> hcmod(x * hcnj y)"
paulson@14469
   773
apply (cases x, cases y)
paulson@14374
   774
apply (simp add: hcmod hcnj hcomplex_mult hRe hypreal_le)
paulson@14314
   775
done
paulson@14314
   776
paulson@14374
   777
lemma hcomplex_hRe_mult_hcnj_le_hcmod2 [simp]: "hRe(x * hcnj y) \<le> hcmod(x * y)"
paulson@14374
   778
apply (cut_tac x = x and y = y in hcomplex_hRe_mult_hcnj_le_hcmod)
paulson@14314
   779
apply (simp add: hcmod_mult)
paulson@14314
   780
done
paulson@14314
   781
paulson@14374
   782
lemma hcmod_triangle_squared [simp]: "hcmod (x + y) ^ 2 \<le> (hcmod(x) + hcmod(y)) ^ 2"
paulson@14469
   783
apply (cases x, cases y)
paulson@14374
   784
apply (simp add: hcmod hcnj hcomplex_add hypreal_mult hypreal_add
paulson@14323
   785
                      hypreal_le realpow_two [symmetric] numeral_2_eq_2
paulson@14374
   786
            del: realpow_Suc)
paulson@14374
   787
apply (simp add: numeral_2_eq_2 [symmetric])
paulson@14314
   788
done
paulson@14314
   789
paulson@14374
   790
lemma hcmod_triangle_ineq [simp]: "hcmod (x + y) \<le> hcmod(x) + hcmod(y)"
paulson@14469
   791
apply (cases x, cases y)
paulson@14374
   792
apply (simp add: hcmod hcomplex_add hypreal_add hypreal_le)
paulson@14314
   793
done
paulson@14314
   794
paulson@14374
   795
lemma hcmod_triangle_ineq2 [simp]: "hcmod(b + a) - hcmod b \<le> hcmod a"
paulson@14374
   796
apply (cut_tac x1 = b and y1 = a and c = "-hcmod b" in hcmod_triangle_ineq [THEN add_right_mono])
paulson@14331
   797
apply (simp add: add_ac)
paulson@14314
   798
done
paulson@14314
   799
paulson@14314
   800
lemma hcmod_diff_commute: "hcmod (x - y) = hcmod (y - x)"
paulson@14469
   801
apply (cases x, cases y)
paulson@14374
   802
apply (simp add: hcmod hcomplex_diff complex_mod_diff_commute)
paulson@14314
   803
done
paulson@14314
   804
paulson@14335
   805
lemma hcmod_add_less:
paulson@14335
   806
     "[| hcmod x < r; hcmod y < s |] ==> hcmod (x + y) < r + s"
paulson@14469
   807
apply (cases x, cases y, cases r, cases s)
paulson@14374
   808
apply (simp add: hcmod hcomplex_add hypreal_add hypreal_less, ultra)
paulson@14314
   809
apply (auto intro: complex_mod_add_less)
paulson@14314
   810
done
paulson@14314
   811
paulson@14335
   812
lemma hcmod_mult_less:
paulson@14335
   813
     "[| hcmod x < r; hcmod y < s |] ==> hcmod (x * y) < r * s"
paulson@14469
   814
apply (cases x, cases y, cases r, cases s)
paulson@14374
   815
apply (simp add: hcmod hypreal_mult hypreal_less hcomplex_mult, ultra)
paulson@14314
   816
apply (auto intro: complex_mod_mult_less)
paulson@14314
   817
done
paulson@14314
   818
paulson@14374
   819
lemma hcmod_diff_ineq [simp]: "hcmod(a) - hcmod(b) \<le> hcmod(a + b)"
paulson@14469
   820
apply (cases a, cases b)
paulson@14374
   821
apply (simp add: hcmod hcomplex_add hypreal_diff hypreal_le)
paulson@14314
   822
done
paulson@14314
   823
paulson@14314
   824
paulson@14314
   825
subsection{*A Few Nonlinear Theorems*}
paulson@14314
   826
paulson@14314
   827
lemma hcpow:
paulson@14314
   828
  "Abs_hcomplex(hcomplexrel``{%n. X n}) hcpow
paulson@14314
   829
   Abs_hypnat(hypnatrel``{%n. Y n}) =
paulson@14314
   830
   Abs_hcomplex(hcomplexrel``{%n. X n ^ Y n})"
paulson@14374
   831
apply (simp add: hcpow_def)
paulson@14374
   832
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   833
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   834
done
paulson@14314
   835
paulson@14335
   836
lemma hcomplex_of_hypreal_hyperpow:
paulson@14335
   837
     "hcomplex_of_hypreal (x pow n) = (hcomplex_of_hypreal x) hcpow n"
paulson@14469
   838
apply (cases x, cases n)
paulson@14374
   839
apply (simp add: hcomplex_of_hypreal hyperpow hcpow complex_of_real_pow)
paulson@14314
   840
done
paulson@14314
   841
paulson@14314
   842
lemma hcmod_hcpow: "hcmod(x hcpow n) = hcmod(x) pow n"
paulson@14469
   843
apply (cases x, cases n)
paulson@14374
   844
apply (simp add: hcpow hyperpow hcmod complex_mod_complexpow)
paulson@14314
   845
done
paulson@14314
   846
paulson@14314
   847
lemma hcmod_hcomplex_inverse: "hcmod(inverse x) = inverse(hcmod x)"
paulson@14374
   848
apply (case_tac "x = 0", simp)
paulson@14314
   849
apply (rule_tac c1 = "hcmod x" in hypreal_mult_left_cancel [THEN iffD1])
paulson@14314
   850
apply (auto simp add: hcmod_mult [symmetric])
paulson@14314
   851
done
paulson@14314
   852
paulson@14374
   853
lemma hcmod_divide: "hcmod(x/y) = hcmod(x)/(hcmod y)"
paulson@14374
   854
by (simp add: hcomplex_divide_def hypreal_divide_def hcmod_mult hcmod_hcomplex_inverse)
paulson@14314
   855
paulson@14354
   856
paulson@14354
   857
subsection{*Exponentiation*}
paulson@14354
   858
paulson@14354
   859
primrec
paulson@14354
   860
     hcomplexpow_0:   "z ^ 0       = 1"
paulson@14354
   861
     hcomplexpow_Suc: "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
paulson@14354
   862
paulson@15003
   863
instance hcomplex :: recpower
paulson@14354
   864
proof
paulson@14354
   865
  fix z :: hcomplex
paulson@14354
   866
  fix n :: nat
paulson@14354
   867
  show "z^0 = 1" by simp
paulson@14354
   868
  show "z^(Suc n) = z * (z^n)" by simp
paulson@14354
   869
qed
paulson@14354
   870
paulson@14377
   871
lemma hcomplexpow_i_squared [simp]: "iii ^ 2 = - 1"
paulson@14377
   872
by (simp add: power2_eq_square)
paulson@14377
   873
paulson@14354
   874
paulson@14354
   875
lemma hcomplex_of_hypreal_pow:
paulson@14354
   876
     "hcomplex_of_hypreal (x ^ n) = (hcomplex_of_hypreal x) ^ n"
paulson@14354
   877
apply (induct_tac "n")
paulson@14354
   878
apply (auto simp add: hcomplex_of_hypreal_mult [symmetric])
paulson@14354
   879
done
paulson@14354
   880
paulson@14354
   881
lemma hcomplex_hcnj_pow: "hcnj(z ^ n) = hcnj(z) ^ n"
paulson@14314
   882
apply (induct_tac "n")
paulson@14354
   883
apply (auto simp add: hcomplex_hcnj_mult)
paulson@14354
   884
done
paulson@14354
   885
paulson@14354
   886
lemma hcmod_hcomplexpow: "hcmod(x ^ n) = hcmod(x) ^ n"
paulson@14354
   887
apply (induct_tac "n")
paulson@14354
   888
apply (auto simp add: hcmod_mult)
paulson@14354
   889
done
paulson@14354
   890
paulson@14354
   891
lemma hcpow_minus:
paulson@14354
   892
     "(-x::hcomplex) hcpow n =
paulson@14354
   893
      (if ( *pNat* even) n then (x hcpow n) else -(x hcpow n))"
paulson@14469
   894
apply (cases x, cases n)
paulson@14374
   895
apply (auto simp add: hcpow hyperpow starPNat hcomplex_minus, ultra)
paulson@14443
   896
apply (auto simp add: neg_power_if, ultra)
paulson@14314
   897
done
paulson@14314
   898
paulson@14314
   899
lemma hcpow_mult: "((r::hcomplex) * s) hcpow n = (r hcpow n) * (s hcpow n)"
paulson@14469
   900
apply (cases r, cases s, cases n)
paulson@14374
   901
apply (simp add: hcpow hypreal_mult hcomplex_mult power_mult_distrib)
paulson@14314
   902
done
paulson@14314
   903
paulson@14354
   904
lemma hcpow_zero [simp]: "0 hcpow (n + 1) = 0"
paulson@14469
   905
apply (simp add: hcomplex_zero_def hypnat_one_def, cases n)
paulson@14374
   906
apply (simp add: hcpow hypnat_add)
paulson@14314
   907
done
paulson@14314
   908
paulson@14354
   909
lemma hcpow_zero2 [simp]: "0 hcpow (hSuc n) = 0"
paulson@14374
   910
by (simp add: hSuc_def)
paulson@14314
   911
paulson@14354
   912
lemma hcpow_not_zero [simp,intro]: "r \<noteq> 0 ==> r hcpow n \<noteq> (0::hcomplex)"
paulson@14469
   913
apply (cases r, cases n)
paulson@14374
   914
apply (auto simp add: hcpow hcomplex_zero_def, ultra)
paulson@14314
   915
done
paulson@14314
   916
paulson@14314
   917
lemma hcpow_zero_zero: "r hcpow n = (0::hcomplex) ==> r = 0"
paulson@14374
   918
by (blast intro: ccontr dest: hcpow_not_zero)
paulson@14314
   919
paulson@14314
   920
lemma hcomplex_divide:
paulson@14314
   921
  "Abs_hcomplex(hcomplexrel``{%n. X n}) / Abs_hcomplex(hcomplexrel``{%n. Y n}) =
paulson@14314
   922
   Abs_hcomplex(hcomplexrel``{%n. X n / Y n})"
paulson@14374
   923
by (simp add: hcomplex_divide_def complex_divide_def hcomplex_inverse hcomplex_mult)
paulson@14374
   924
paulson@14314
   925
paulson@14314
   926
paulson@14377
   927
paulson@14314
   928
subsection{*The Function @{term hsgn}*}
paulson@14314
   929
paulson@14314
   930
lemma hsgn:
paulson@14314
   931
  "hsgn (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
   932
      Abs_hcomplex(hcomplexrel `` {%n. sgn (X n)})"
paulson@14374
   933
apply (simp add: hsgn_def)
paulson@14374
   934
apply (rule_tac f = Abs_hcomplex in arg_cong)
paulson@14377
   935
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
   936
done
paulson@14314
   937
paulson@14374
   938
lemma hsgn_zero [simp]: "hsgn 0 = 0"
paulson@14374
   939
by (simp add: hcomplex_zero_def hsgn)
paulson@14314
   940
paulson@14374
   941
lemma hsgn_one [simp]: "hsgn 1 = 1"
paulson@14374
   942
by (simp add: hcomplex_one_def hsgn)
paulson@14314
   943
paulson@14314
   944
lemma hsgn_minus: "hsgn (-z) = - hsgn(z)"
paulson@14469
   945
apply (cases z)
paulson@14374
   946
apply (simp add: hsgn hcomplex_minus sgn_minus)
paulson@14314
   947
done
paulson@14314
   948
paulson@14314
   949
lemma hsgn_eq: "hsgn z = z / hcomplex_of_hypreal (hcmod z)"
paulson@14469
   950
apply (cases z)
paulson@14374
   951
apply (simp add: hsgn hcomplex_divide hcomplex_of_hypreal hcmod sgn_eq)
paulson@14314
   952
done
paulson@14314
   953
paulson@14314
   954
paulson@14377
   955
lemma hcmod_i: "hcmod (HComplex x y) = ( *f* sqrt) (x ^ 2 + y ^ 2)"
paulson@14469
   956
apply (cases x, cases y) 
paulson@14377
   957
apply (simp add: HComplex_eq_Abs_hcomplex_Complex starfun 
paulson@14377
   958
                 hypreal_mult hypreal_add hcmod numeral_2_eq_2)
paulson@14314
   959
done
paulson@14314
   960
paulson@14377
   961
lemma hcomplex_eq_cancel_iff1 [simp]:
paulson@14377
   962
     "(hcomplex_of_hypreal xa = HComplex x y) = (xa = x & y = 0)"
paulson@14377
   963
by (simp add: hcomplex_of_hypreal_eq)
paulson@14314
   964
paulson@14374
   965
lemma hcomplex_eq_cancel_iff2 [simp]:
paulson@14377
   966
     "(HComplex x y = hcomplex_of_hypreal xa) = (x = xa & y = 0)"
paulson@14377
   967
by (simp add: hcomplex_of_hypreal_eq)
paulson@14314
   968
paulson@14377
   969
lemma HComplex_eq_0 [simp]: "(HComplex x y = 0) = (x = 0 & y = 0)"
paulson@14377
   970
by (insert hcomplex_eq_cancel_iff2 [of _ _ 0], simp)
paulson@14314
   971
paulson@14377
   972
lemma HComplex_eq_1 [simp]: "(HComplex x y = 1) = (x = 1 & y = 0)"
paulson@14377
   973
by (insert hcomplex_eq_cancel_iff2 [of _ _ 1], simp)
paulson@14314
   974
paulson@14377
   975
lemma i_eq_HComplex_0_1: "iii = HComplex 0 1"
paulson@14377
   976
by (insert hcomplex_of_hypreal_i [of 1], simp)
paulson@14314
   977
paulson@14377
   978
lemma HComplex_eq_i [simp]: "(HComplex x y = iii) = (x = 0 & y = 1)"
paulson@14377
   979
by (simp add: i_eq_HComplex_0_1) 
paulson@14314
   980
paulson@14374
   981
lemma hRe_hsgn [simp]: "hRe(hsgn z) = hRe(z)/hcmod z"
paulson@14469
   982
apply (cases z)
paulson@14374
   983
apply (simp add: hsgn hcmod hRe hypreal_divide)
paulson@14314
   984
done
paulson@14314
   985
paulson@14374
   986
lemma hIm_hsgn [simp]: "hIm(hsgn z) = hIm(z)/hcmod z"
paulson@14469
   987
apply (cases z)
paulson@14374
   988
apply (simp add: hsgn hcmod hIm hypreal_divide)
paulson@14314
   989
done
paulson@14314
   990
paulson@14374
   991
lemma real_two_squares_add_zero_iff [simp]: "(x*x + y*y = 0) = ((x::real) = 0 & y = 0)"
paulson@14377
   992
by (auto intro: real_sum_squares_cancel)
paulson@14314
   993
paulson@14335
   994
lemma hcomplex_inverse_complex_split:
paulson@14335
   995
     "inverse(hcomplex_of_hypreal x + iii * hcomplex_of_hypreal y) =
paulson@14314
   996
      hcomplex_of_hypreal(x/(x ^ 2 + y ^ 2)) -
paulson@14314
   997
      iii * hcomplex_of_hypreal(y/(x ^ 2 + y ^ 2))"
paulson@14469
   998
apply (cases x, cases y)
paulson@15013
   999
apply (simp add: hcomplex_of_hypreal hcomplex_mult hcomplex_add iii_def
paulson@15013
  1000
         starfun hypreal_mult hypreal_add hcomplex_inverse hypreal_divide
paulson@15013
  1001
         hcomplex_diff numeral_2_eq_2 complex_of_real_def i_def)
paulson@14377
  1002
apply (simp add: diff_minus) 
paulson@14374
  1003
done
paulson@14374
  1004
paulson@14377
  1005
lemma HComplex_inverse:
paulson@14377
  1006
     "inverse (HComplex x y) =
paulson@14377
  1007
      HComplex (x/(x ^ 2 + y ^ 2)) (-y/(x ^ 2 + y ^ 2))"
paulson@14377
  1008
by (simp only: HComplex_def hcomplex_inverse_complex_split, simp)
paulson@14377
  1009
paulson@14377
  1010
paulson@14377
  1011
paulson@14374
  1012
lemma hRe_mult_i_eq[simp]:
paulson@14374
  1013
    "hRe (iii * hcomplex_of_hypreal y) = 0"
paulson@14469
  1014
apply (simp add: iii_def, cases y)
paulson@14374
  1015
apply (simp add: hcomplex_of_hypreal hcomplex_mult hRe hypreal_zero_num)
paulson@14314
  1016
done
paulson@14314
  1017
paulson@14374
  1018
lemma hIm_mult_i_eq [simp]:
paulson@14314
  1019
    "hIm (iii * hcomplex_of_hypreal y) = y"
paulson@14469
  1020
apply (simp add: iii_def, cases y)
paulson@14374
  1021
apply (simp add: hcomplex_of_hypreal hcomplex_mult hIm hypreal_zero_num)
paulson@14314
  1022
done
paulson@14314
  1023
paulson@14374
  1024
lemma hcmod_mult_i [simp]: "hcmod (iii * hcomplex_of_hypreal y) = abs y"
paulson@14469
  1025
apply (cases y)
paulson@14374
  1026
apply (simp add: hcomplex_of_hypreal hcmod hypreal_hrabs iii_def hcomplex_mult)
paulson@14314
  1027
done
paulson@14314
  1028
paulson@14374
  1029
lemma hcmod_mult_i2 [simp]: "hcmod (hcomplex_of_hypreal y * iii) = abs y"
paulson@14377
  1030
by (simp only: hcmod_mult_i hcomplex_mult_commute)
paulson@14314
  1031
paulson@14314
  1032
(*---------------------------------------------------------------------------*)
paulson@14314
  1033
(*  harg                                                                     *)
paulson@14314
  1034
(*---------------------------------------------------------------------------*)
paulson@14314
  1035
paulson@14314
  1036
lemma harg:
paulson@14314
  1037
  "harg (Abs_hcomplex(hcomplexrel `` {%n. X n})) =
paulson@14314
  1038
      Abs_hypreal(hyprel `` {%n. arg (X n)})"
paulson@14374
  1039
apply (simp add: harg_def)
paulson@14374
  1040
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14377
  1041
apply (auto iff: hcomplexrel_iff, ultra)
paulson@14314
  1042
done
paulson@14314
  1043
paulson@14354
  1044
lemma cos_harg_i_mult_zero_pos:
paulson@14377
  1045
     "0 < y ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
paulson@14469
  1046
apply (cases y)
paulson@14377
  1047
apply (simp add: HComplex_def hcomplex_of_hypreal iii_def hcomplex_mult
paulson@14377
  1048
                hcomplex_add hypreal_zero_num hypreal_less starfun harg, ultra)
paulson@14314
  1049
done
paulson@14314
  1050
paulson@14354
  1051
lemma cos_harg_i_mult_zero_neg:
paulson@14377
  1052
     "y < 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
paulson@14469
  1053
apply (cases y)
paulson@14377
  1054
apply (simp add: HComplex_def hcomplex_of_hypreal iii_def hcomplex_mult
paulson@14377
  1055
                 hcomplex_add hypreal_zero_num hypreal_less starfun harg, ultra)
paulson@14314
  1056
done
paulson@14314
  1057
paulson@14354
  1058
lemma cos_harg_i_mult_zero [simp]:
paulson@14377
  1059
     "y \<noteq> 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
paulson@14377
  1060
by (auto simp add: linorder_neq_iff
paulson@14377
  1061
                   cos_harg_i_mult_zero_pos cos_harg_i_mult_zero_neg)
paulson@14354
  1062
paulson@14354
  1063
lemma hcomplex_of_hypreal_zero_iff [simp]:
paulson@14354
  1064
     "(hcomplex_of_hypreal y = 0) = (y = 0)"
paulson@14469
  1065
apply (cases y)
paulson@14374
  1066
apply (simp add: hcomplex_of_hypreal hypreal_zero_num hcomplex_zero_def)
paulson@14314
  1067
done
paulson@14314
  1068
paulson@14314
  1069
paulson@14354
  1070
subsection{*Polar Form for Nonstandard Complex Numbers*}
paulson@14314
  1071
paulson@14335
  1072
lemma complex_split_polar2:
paulson@14377
  1073
     "\<forall>n. \<exists>r a. (z n) =  complex_of_real r * (Complex (cos a) (sin a))"
paulson@14377
  1074
by (blast intro: complex_split_polar)
paulson@14377
  1075
paulson@14377
  1076
lemma lemma_hypreal_P_EX2:
paulson@14377
  1077
     "(\<exists>(x::hypreal) y. P x y) =
paulson@14377
  1078
      (\<exists>f g. P (Abs_hypreal(hyprel `` {f})) (Abs_hypreal(hyprel `` {g})))"
paulson@14377
  1079
apply auto
paulson@14377
  1080
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14377
  1081
apply (rule_tac z = y in eq_Abs_hypreal, auto)
paulson@14314
  1082
done
paulson@14314
  1083
paulson@14314
  1084
lemma hcomplex_split_polar:
paulson@14377
  1085
  "\<exists>r a. z = hcomplex_of_hypreal r * (HComplex(( *f* cos) a)(( *f* sin) a))"
paulson@14469
  1086
apply (cases z)
paulson@14377
  1087
apply (simp add: lemma_hypreal_P_EX2 hcomplex_of_hypreal iii_def starfun hcomplex_add hcomplex_mult HComplex_def)
paulson@14374
  1088
apply (cut_tac z = x in complex_split_polar2)
paulson@14335
  1089
apply (drule choice, safe)+
paulson@14374
  1090
apply (rule_tac x = f in exI)
paulson@14374
  1091
apply (rule_tac x = fa in exI, auto)
paulson@14314
  1092
done
paulson@14314
  1093
paulson@14314
  1094
lemma hcis:
paulson@14314
  1095
  "hcis (Abs_hypreal(hyprel `` {%n. X n})) =
paulson@14314
  1096
      Abs_hcomplex(hcomplexrel `` {%n. cis (X n)})"
paulson@14374
  1097
apply (simp add: hcis_def)
paulson@14377
  1098
apply (rule_tac f = Abs_hcomplex in arg_cong, auto iff: hcomplexrel_iff, ultra)
paulson@14314
  1099
done
paulson@14314
  1100
paulson@14314
  1101
lemma hcis_eq:
paulson@14314
  1102
   "hcis a =
paulson@14314
  1103
    (hcomplex_of_hypreal(( *f* cos) a) +
paulson@14314
  1104
    iii * hcomplex_of_hypreal(( *f* sin) a))"
paulson@14469
  1105
apply (cases a)
paulson@14374
  1106
apply (simp add: starfun hcis hcomplex_of_hypreal iii_def hcomplex_mult hcomplex_add cis_def)
paulson@14314
  1107
done
paulson@14314
  1108
paulson@14314
  1109
lemma hrcis:
paulson@14314
  1110
  "hrcis (Abs_hypreal(hyprel `` {%n. X n})) (Abs_hypreal(hyprel `` {%n. Y n})) =
paulson@14314
  1111
      Abs_hcomplex(hcomplexrel `` {%n. rcis (X n) (Y n)})"
paulson@14374
  1112
by (simp add: hrcis_def hcomplex_of_hypreal hcomplex_mult hcis rcis_def)
paulson@14314
  1113
paulson@14354
  1114
lemma hrcis_Ex: "\<exists>r a. z = hrcis r a"
paulson@14377
  1115
apply (simp add: hrcis_def hcis_eq hcomplex_of_hypreal_mult_HComplex [symmetric])
paulson@14314
  1116
apply (rule hcomplex_split_polar)
paulson@14314
  1117
done
paulson@14314
  1118
paulson@14374
  1119
lemma hRe_hcomplex_polar [simp]:
paulson@14377
  1120
     "hRe (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = 
paulson@14377
  1121
      r * ( *f* cos) a"
paulson@14377
  1122
by (simp add: hcomplex_of_hypreal_mult_HComplex)
paulson@14314
  1123
paulson@14374
  1124
lemma hRe_hrcis [simp]: "hRe(hrcis r a) = r * ( *f* cos) a"
paulson@14374
  1125
by (simp add: hrcis_def hcis_eq)
paulson@14314
  1126
paulson@14374
  1127
lemma hIm_hcomplex_polar [simp]:
paulson@14377
  1128
     "hIm (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = 
paulson@14377
  1129
      r * ( *f* sin) a"
paulson@14377
  1130
by (simp add: hcomplex_of_hypreal_mult_HComplex)
paulson@14314
  1131
paulson@14374
  1132
lemma hIm_hrcis [simp]: "hIm(hrcis r a) = r * ( *f* sin) a"
paulson@14374
  1133
by (simp add: hrcis_def hcis_eq)
paulson@14314
  1134
paulson@14377
  1135
paulson@14377
  1136
lemma hcmod_unit_one [simp]:
paulson@14377
  1137
     "hcmod (HComplex (( *f* cos) a) (( *f* sin) a)) = 1"
paulson@14469
  1138
apply (cases a) 
paulson@14377
  1139
apply (simp add: HComplex_def iii_def starfun hcomplex_of_hypreal 
paulson@14377
  1140
                 hcomplex_mult hcmod hcomplex_add hypreal_one_def)
paulson@14377
  1141
done
paulson@14377
  1142
paulson@14374
  1143
lemma hcmod_complex_polar [simp]:
paulson@14377
  1144
     "hcmod (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) =
paulson@14377
  1145
      abs r"
paulson@14377
  1146
apply (simp only: hcmod_mult hcmod_unit_one, simp)  
paulson@14314
  1147
done
paulson@14314
  1148
paulson@14374
  1149
lemma hcmod_hrcis [simp]: "hcmod(hrcis r a) = abs r"
paulson@14374
  1150
by (simp add: hrcis_def hcis_eq)
paulson@14314
  1151
paulson@14314
  1152
(*---------------------------------------------------------------------------*)
paulson@14314
  1153
(*  (r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b)                *)
paulson@14314
  1154
(*---------------------------------------------------------------------------*)
paulson@14314
  1155
paulson@14314
  1156
lemma hcis_hrcis_eq: "hcis a = hrcis 1 a"
paulson@14374
  1157
by (simp add: hrcis_def)
paulson@14314
  1158
declare hcis_hrcis_eq [symmetric, simp]
paulson@14314
  1159
paulson@14314
  1160
lemma hrcis_mult:
paulson@14314
  1161
  "hrcis r1 a * hrcis r2 b = hrcis (r1*r2) (a + b)"
paulson@14469
  1162
apply (simp add: hrcis_def, cases r1, cases r2, cases a, cases b)
paulson@14374
  1163
apply (simp add: hrcis hcis hypreal_add hypreal_mult hcomplex_of_hypreal
paulson@15013
  1164
                      hcomplex_mult cis_mult [symmetric])
paulson@14314
  1165
done
paulson@14314
  1166
paulson@14314
  1167
lemma hcis_mult: "hcis a * hcis b = hcis (a + b)"
paulson@14469
  1168
apply (cases a, cases b)
paulson@14374
  1169
apply (simp add: hcis hcomplex_mult hypreal_add cis_mult)
paulson@14314
  1170
done
paulson@14314
  1171
paulson@14374
  1172
lemma hcis_zero [simp]: "hcis 0 = 1"
paulson@14374
  1173
by (simp add: hcomplex_one_def hcis hypreal_zero_num)
paulson@14314
  1174
paulson@14374
  1175
lemma hrcis_zero_mod [simp]: "hrcis 0 a = 0"
paulson@14469
  1176
apply (simp add: hcomplex_zero_def, cases a)
paulson@14374
  1177
apply (simp add: hrcis hypreal_zero_num)
paulson@14314
  1178
done
paulson@14314
  1179
paulson@14374
  1180
lemma hrcis_zero_arg [simp]: "hrcis r 0 = hcomplex_of_hypreal r"
paulson@14469
  1181
apply (cases r)
paulson@14374
  1182
apply (simp add: hrcis hypreal_zero_num hcomplex_of_hypreal)
paulson@14314
  1183
done
paulson@14314
  1184
paulson@14374
  1185
lemma hcomplex_i_mult_minus [simp]: "iii * (iii * x) = - x"
paulson@14374
  1186
by (simp add: hcomplex_mult_assoc [symmetric])
paulson@14314
  1187
paulson@14374
  1188
lemma hcomplex_i_mult_minus2 [simp]: "iii * iii * x = - x"
paulson@14374
  1189
by simp
paulson@14314
  1190
paulson@14314
  1191
lemma hcis_hypreal_of_nat_Suc_mult:
paulson@14314
  1192
   "hcis (hypreal_of_nat (Suc n) * a) = hcis a * hcis (hypreal_of_nat n * a)"
paulson@14469
  1193
apply (cases a)
paulson@14374
  1194
apply (simp add: hypreal_of_nat hcis hypreal_mult hcomplex_mult cis_real_of_nat_Suc_mult)
paulson@14314
  1195
done
paulson@14314
  1196
paulson@14314
  1197
lemma NSDeMoivre: "(hcis a) ^ n = hcis (hypreal_of_nat n * a)"
paulson@14314
  1198
apply (induct_tac "n")
paulson@14374
  1199
apply (simp_all add: hcis_hypreal_of_nat_Suc_mult)
paulson@14314
  1200
done
paulson@14314
  1201
paulson@14335
  1202
lemma hcis_hypreal_of_hypnat_Suc_mult:
paulson@14335
  1203
     "hcis (hypreal_of_hypnat (n + 1) * a) =
paulson@14314
  1204
      hcis a * hcis (hypreal_of_hypnat n * a)"
paulson@14469
  1205
apply (cases a, cases n)
paulson@14374
  1206
apply (simp add: hcis hypreal_of_hypnat hypnat_add hypnat_one_def hypreal_mult hcomplex_mult cis_real_of_nat_Suc_mult)
paulson@14314
  1207
done
paulson@14314
  1208
paulson@14314
  1209
lemma NSDeMoivre_ext: "(hcis a) hcpow n = hcis (hypreal_of_hypnat n * a)"
paulson@14469
  1210
apply (cases a, cases n)
paulson@14374
  1211
apply (simp add: hcis hypreal_of_hypnat hypreal_mult hcpow DeMoivre)
paulson@14314
  1212
done
paulson@14314
  1213
paulson@14314
  1214
lemma DeMoivre2:
paulson@14314
  1215
  "(hrcis r a) ^ n = hrcis (r ^ n) (hypreal_of_nat n * a)"
paulson@14374
  1216
apply (simp add: hrcis_def power_mult_distrib NSDeMoivre hcomplex_of_hypreal_pow)
paulson@14314
  1217
done
paulson@14314
  1218
paulson@14314
  1219
lemma DeMoivre2_ext:
paulson@14314
  1220
  "(hrcis r a) hcpow n = hrcis (r pow n) (hypreal_of_hypnat n * a)"
paulson@14374
  1221
apply (simp add: hrcis_def hcpow_mult NSDeMoivre_ext hcomplex_of_hypreal_hyperpow)
paulson@14374
  1222
done
paulson@14374
  1223
paulson@14374
  1224
lemma hcis_inverse [simp]: "inverse(hcis a) = hcis (-a)"
paulson@14469
  1225
apply (cases a)
paulson@14374
  1226
apply (simp add: hcomplex_inverse hcis hypreal_minus)
paulson@14314
  1227
done
paulson@14314
  1228
paulson@14374
  1229
lemma hrcis_inverse: "inverse(hrcis r a) = hrcis (inverse r) (-a)"
paulson@14469
  1230
apply (cases a, cases r)
paulson@14374
  1231
apply (simp add: hcomplex_inverse hrcis hypreal_minus hypreal_inverse rcis_inverse, ultra)
paulson@14374
  1232
apply (simp add: real_divide_def)
paulson@14314
  1233
done
paulson@14314
  1234
paulson@14374
  1235
lemma hRe_hcis [simp]: "hRe(hcis a) = ( *f* cos) a"
paulson@14469
  1236
apply (cases a)
paulson@14374
  1237
apply (simp add: hcis starfun hRe)
paulson@14314
  1238
done
paulson@14314
  1239
paulson@14374
  1240
lemma hIm_hcis [simp]: "hIm(hcis a) = ( *f* sin) a"
paulson@14469
  1241
apply (cases a)
paulson@14374
  1242
apply (simp add: hcis starfun hIm)
paulson@14314
  1243
done
paulson@14314
  1244
paulson@14374
  1245
lemma cos_n_hRe_hcis_pow_n: "( *f* cos) (hypreal_of_nat n * a) = hRe(hcis a ^ n)"
paulson@14377
  1246
by (simp add: NSDeMoivre)
paulson@14314
  1247
paulson@14374
  1248
lemma sin_n_hIm_hcis_pow_n: "( *f* sin) (hypreal_of_nat n * a) = hIm(hcis a ^ n)"
paulson@14377
  1249
by (simp add: NSDeMoivre)
paulson@14314
  1250
paulson@14374
  1251
lemma cos_n_hRe_hcis_hcpow_n: "( *f* cos) (hypreal_of_hypnat n * a) = hRe(hcis a hcpow n)"
paulson@14377
  1252
by (simp add: NSDeMoivre_ext)
paulson@14314
  1253
paulson@14374
  1254
lemma sin_n_hIm_hcis_hcpow_n: "( *f* sin) (hypreal_of_hypnat n * a) = hIm(hcis a hcpow n)"
paulson@14377
  1255
by (simp add: NSDeMoivre_ext)
paulson@14314
  1256
paulson@14314
  1257
lemma hexpi_add: "hexpi(a + b) = hexpi(a) * hexpi(b)"
paulson@14469
  1258
apply (simp add: hexpi_def, cases a, cases b)
paulson@14374
  1259
apply (simp add: hcis hRe hIm hcomplex_add hcomplex_mult hypreal_mult starfun hcomplex_of_hypreal cis_mult [symmetric] complex_Im_add complex_Re_add exp_add complex_of_real_mult)
paulson@14314
  1260
done
paulson@14314
  1261
paulson@14314
  1262
paulson@14374
  1263
subsection{*@{term hcomplex_of_complex}: the Injection from
paulson@14354
  1264
  type @{typ complex} to to @{typ hcomplex}*}
paulson@14354
  1265
paulson@14354
  1266
lemma inj_hcomplex_of_complex: "inj(hcomplex_of_complex)"
paulson@14374
  1267
apply (rule inj_onI, rule ccontr)
paulson@14374
  1268
apply (simp add: hcomplex_of_complex_def)
paulson@14354
  1269
done
paulson@14354
  1270
paulson@14354
  1271
lemma hcomplex_of_complex_i: "iii = hcomplex_of_complex ii"
paulson@14374
  1272
by (simp add: iii_def hcomplex_of_complex_def)
paulson@14314
  1273
paulson@14374
  1274
lemma hcomplex_of_complex_add [simp]:
paulson@14314
  1275
     "hcomplex_of_complex (z1 + z2) = hcomplex_of_complex z1 + hcomplex_of_complex z2"
paulson@14374
  1276
by (simp add: hcomplex_of_complex_def hcomplex_add)
paulson@14314
  1277
paulson@14374
  1278
lemma hcomplex_of_complex_mult [simp]:
paulson@14314
  1279
     "hcomplex_of_complex (z1 * z2) = hcomplex_of_complex z1 * hcomplex_of_complex z2"
paulson@14374
  1280
by (simp add: hcomplex_of_complex_def hcomplex_mult)
paulson@14314
  1281
paulson@14374
  1282
lemma hcomplex_of_complex_eq_iff [simp]:
paulson@14374
  1283
     "(hcomplex_of_complex z1 = hcomplex_of_complex z2) = (z1 = z2)"
paulson@14374
  1284
by (simp add: hcomplex_of_complex_def)
paulson@14314
  1285
paulson@14374
  1286
paulson@14374
  1287
lemma hcomplex_of_complex_minus [simp]:
paulson@14335
  1288
     "hcomplex_of_complex (-r) = - hcomplex_of_complex  r"
paulson@14374
  1289
by (simp add: hcomplex_of_complex_def hcomplex_minus)
paulson@14314
  1290
paulson@14374
  1291
lemma hcomplex_of_complex_one [simp]: "hcomplex_of_complex 1 = 1"
paulson@14374
  1292
by (simp add: hcomplex_of_complex_def hcomplex_one_def)
paulson@14314
  1293
paulson@14374
  1294
lemma hcomplex_of_complex_zero [simp]: "hcomplex_of_complex 0 = 0"
paulson@14374
  1295
by (simp add: hcomplex_of_complex_def hcomplex_zero_def)
paulson@14314
  1296
paulson@14387
  1297
lemma hcomplex_of_complex_zero_iff [simp]:
paulson@14387
  1298
     "(hcomplex_of_complex r = 0) = (r = 0)"
paulson@14387
  1299
by (auto intro: FreeUltrafilterNat_P 
paulson@14387
  1300
         simp add: hcomplex_of_complex_def hcomplex_zero_def)
paulson@14314
  1301
paulson@14374
  1302
lemma hcomplex_of_complex_inverse [simp]:
paulson@14335
  1303
     "hcomplex_of_complex (inverse r) = inverse (hcomplex_of_complex r)"
paulson@15013
  1304
proof cases
paulson@15013
  1305
  assume "r=0" thus ?thesis by simp
paulson@15013
  1306
next
paulson@15013
  1307
  assume nz: "r\<noteq>0" 
paulson@15013
  1308
  show ?thesis
paulson@15013
  1309
  proof (rule hcomplex_mult_left_cancel [THEN iffD1]) 
paulson@15013
  1310
    show "hcomplex_of_complex r \<noteq> 0"
paulson@15013
  1311
      by (simp add: nz) 
paulson@15013
  1312
  next
paulson@15013
  1313
    have "hcomplex_of_complex r * hcomplex_of_complex (inverse r) =
paulson@15013
  1314
          hcomplex_of_complex (r * inverse r)"
paulson@15013
  1315
      by simp
paulson@15013
  1316
    also have "... = hcomplex_of_complex r * inverse (hcomplex_of_complex r)" 
paulson@15013
  1317
      by (simp add: nz)
paulson@15013
  1318
    finally show "hcomplex_of_complex r * hcomplex_of_complex (inverse r) =
paulson@15013
  1319
                  hcomplex_of_complex r * inverse (hcomplex_of_complex r)" .
paulson@15013
  1320
  qed
paulson@15013
  1321
qed
paulson@14314
  1322
paulson@14374
  1323
lemma hcomplex_of_complex_divide [simp]:
paulson@15013
  1324
     "hcomplex_of_complex (z1 / z2) = 
paulson@15013
  1325
      hcomplex_of_complex z1 / hcomplex_of_complex z2"
paulson@14374
  1326
by (simp add: hcomplex_divide_def complex_divide_def)
paulson@14314
  1327
paulson@14314
  1328
lemma hRe_hcomplex_of_complex:
paulson@14314
  1329
   "hRe (hcomplex_of_complex z) = hypreal_of_real (Re z)"
paulson@14374
  1330
by (simp add: hcomplex_of_complex_def hypreal_of_real_def hRe)
paulson@14314
  1331
paulson@14314
  1332
lemma hIm_hcomplex_of_complex:
paulson@14314
  1333
   "hIm (hcomplex_of_complex z) = hypreal_of_real (Im z)"
paulson@14374
  1334
by (simp add: hcomplex_of_complex_def hypreal_of_real_def hIm)
paulson@14314
  1335
paulson@14314
  1336
lemma hcmod_hcomplex_of_complex:
paulson@14314
  1337
     "hcmod (hcomplex_of_complex x) = hypreal_of_real (cmod x)"
paulson@14374
  1338
by (simp add: hypreal_of_real_def hcomplex_of_complex_def hcmod)
paulson@14314
  1339
paulson@14387
  1340
paulson@14387
  1341
subsection{*Numerals and Arithmetic*}
paulson@14387
  1342
paulson@14387
  1343
instance hcomplex :: number ..
paulson@14387
  1344
paulson@15013
  1345
defs (overloaded)
paulson@15013
  1346
  hcomplex_number_of_def: "(number_of w :: hcomplex) == of_int (Rep_Bin w)"
paulson@15013
  1347
    --{*the type constraint is essential!*}
paulson@14387
  1348
paulson@14387
  1349
instance hcomplex :: number_ring
paulson@15013
  1350
by (intro_classes, simp add: hcomplex_number_of_def) 
paulson@15013
  1351
paulson@15013
  1352
paulson@15013
  1353
lemma hcomplex_of_complex_of_nat [simp]:
paulson@15013
  1354
     "hcomplex_of_complex (of_nat n) = of_nat n"
paulson@15013
  1355
by (induct n, simp_all) 
paulson@15013
  1356
paulson@15013
  1357
lemma hcomplex_of_complex_of_int [simp]:
paulson@15013
  1358
     "hcomplex_of_complex (of_int z) = of_int z"
paulson@15013
  1359
proof (cases z)
paulson@15013
  1360
  case (1 n)
paulson@15013
  1361
    thus ?thesis by simp
paulson@15013
  1362
next
paulson@15013
  1363
  case (2 n)
paulson@15013
  1364
    thus ?thesis 
paulson@15013
  1365
      by (simp only: of_int_minus hcomplex_of_complex_minus, simp)
paulson@14387
  1366
qed
paulson@14387
  1367
paulson@14387
  1368
paulson@14387
  1369
text{*Collapse applications of @{term hcomplex_of_complex} to @{term number_of}*}
paulson@14387
  1370
lemma hcomplex_number_of [simp]:
paulson@14387
  1371
     "hcomplex_of_complex (number_of w) = number_of w"
paulson@15013
  1372
by (simp add: hcomplex_number_of_def complex_number_of_def) 
paulson@14387
  1373
paulson@14387
  1374
lemma hcomplex_of_hypreal_eq_hcomplex_of_complex: 
paulson@14387
  1375
     "hcomplex_of_hypreal (hypreal_of_real x) =  
paulson@15013
  1376
      hcomplex_of_complex (complex_of_real x)"
paulson@14387
  1377
by (simp add: hypreal_of_real_def hcomplex_of_hypreal hcomplex_of_complex_def 
paulson@14387
  1378
              complex_of_real_def)
paulson@14387
  1379
paulson@14387
  1380
lemma hcomplex_hypreal_number_of: 
paulson@14387
  1381
  "hcomplex_of_complex (number_of w) = hcomplex_of_hypreal(number_of w)"
paulson@14387
  1382
by (simp only: complex_number_of [symmetric] hypreal_number_of [symmetric] 
paulson@14387
  1383
               hcomplex_of_hypreal_eq_hcomplex_of_complex)
paulson@14387
  1384
paulson@14387
  1385
text{*This theorem is necessary because theorems such as
paulson@14387
  1386
   @{text iszero_number_of_0} only hold for ordered rings. They cannot
paulson@14387
  1387
   be generalized to fields in general because they fail for finite fields.
paulson@14387
  1388
   They work for type complex because the reals can be embedded in them.*}
paulson@14387
  1389
lemma iszero_hcomplex_number_of [simp]:
paulson@14387
  1390
     "iszero (number_of w :: hcomplex) = iszero (number_of w :: real)"
paulson@14387
  1391
apply (simp only: iszero_complex_number_of [symmetric])  
paulson@14387
  1392
apply (simp only: hcomplex_of_complex_zero_iff hcomplex_number_of [symmetric] 
paulson@14387
  1393
                  iszero_def)  
paulson@14387
  1394
done
paulson@14387
  1395
paulson@14387
  1396
paulson@14387
  1397
(*
paulson@14387
  1398
Goal "z + hcnj z =  
paulson@14387
  1399
      hcomplex_of_hypreal (2 * hRe(z))"
paulson@14387
  1400
by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
paulson@14387
  1401
by (auto_tac (claset(),HOL_ss addsimps [hRe,hcnj,hcomplex_add,
paulson@14387
  1402
    hypreal_mult,hcomplex_of_hypreal,complex_add_cnj]));
paulson@14387
  1403
qed "hcomplex_add_hcnj";
paulson@14387
  1404
paulson@14387
  1405
Goal "z - hcnj z = \
paulson@14387
  1406
\     hcomplex_of_hypreal (hypreal_of_real #2 * hIm(z)) * iii";
paulson@14387
  1407
by (res_inst_tac [("z","z")] eq_Abs_hcomplex 1);
paulson@14387
  1408
by (auto_tac (claset(),simpset() addsimps [hIm,hcnj,hcomplex_diff,
paulson@14387
  1409
    hypreal_of_real_def,hypreal_mult,hcomplex_of_hypreal,
paulson@14387
  1410
    complex_diff_cnj,iii_def,hcomplex_mult]));
paulson@14387
  1411
qed "hcomplex_diff_hcnj";
paulson@14387
  1412
*)
paulson@14387
  1413
paulson@14387
  1414
paulson@14387
  1415
lemma hcomplex_hcnj_num_zero_iff: "(hcnj z = 0) = (z = 0)"
paulson@14387
  1416
apply (auto simp add: hcomplex_hcnj_zero_iff)
paulson@14387
  1417
done
paulson@14387
  1418
declare hcomplex_hcnj_num_zero_iff [simp]
paulson@14387
  1419
paulson@14387
  1420
lemma hcomplex_zero_num: "0 = Abs_hcomplex (hcomplexrel `` {%n. 0})"
paulson@14387
  1421
apply (simp add: hcomplex_zero_def)
paulson@14387
  1422
done
paulson@14387
  1423
paulson@14387
  1424
lemma hcomplex_one_num: "1 =  Abs_hcomplex (hcomplexrel `` {%n. 1})"
paulson@14387
  1425
apply (simp add: hcomplex_one_def)
paulson@14387
  1426
done
paulson@14387
  1427
paulson@14387
  1428
(*** Real and imaginary stuff ***)
paulson@14387
  1429
paulson@14387
  1430
(*Convert???
paulson@14387
  1431
Goalw [hcomplex_number_of_def] 
paulson@14387
  1432
  "((number_of xa :: hcomplex) + iii * number_of ya =  
paulson@14387
  1433
        number_of xb + iii * number_of yb) =  
paulson@14387
  1434
   (((number_of xa :: hcomplex) = number_of xb) &  
paulson@14387
  1435
    ((number_of ya :: hcomplex) = number_of yb))"
paulson@14387
  1436
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff,
paulson@14387
  1437
     hcomplex_hypreal_number_of]));
paulson@14387
  1438
qed "hcomplex_number_of_eq_cancel_iff";
paulson@14387
  1439
Addsimps [hcomplex_number_of_eq_cancel_iff];
paulson@14387
  1440
paulson@14387
  1441
Goalw [hcomplex_number_of_def] 
paulson@14387
  1442
  "((number_of xa :: hcomplex) + number_of ya * iii = \
paulson@14387
  1443
\       number_of xb + number_of yb * iii) = \
paulson@14387
  1444
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1445
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1446
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffA,
paulson@14387
  1447
    hcomplex_hypreal_number_of]));
paulson@14387
  1448
qed "hcomplex_number_of_eq_cancel_iffA";
paulson@14387
  1449
Addsimps [hcomplex_number_of_eq_cancel_iffA];
paulson@14387
  1450
paulson@14387
  1451
Goalw [hcomplex_number_of_def] 
paulson@14387
  1452
  "((number_of xa :: hcomplex) + number_of ya * iii = \
paulson@14387
  1453
\       number_of xb + iii * number_of yb) = \
paulson@14387
  1454
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1455
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1456
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffB,
paulson@14387
  1457
    hcomplex_hypreal_number_of]));
paulson@14387
  1458
qed "hcomplex_number_of_eq_cancel_iffB";
paulson@14387
  1459
Addsimps [hcomplex_number_of_eq_cancel_iffB];
paulson@14387
  1460
paulson@14387
  1461
Goalw [hcomplex_number_of_def] 
paulson@14387
  1462
  "((number_of xa :: hcomplex) + iii * number_of ya = \
paulson@14387
  1463
\       number_of xb + number_of yb * iii) = \
paulson@14387
  1464
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1465
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1466
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iffC,
paulson@14387
  1467
     hcomplex_hypreal_number_of]));
paulson@14387
  1468
qed "hcomplex_number_of_eq_cancel_iffC";
paulson@14387
  1469
Addsimps [hcomplex_number_of_eq_cancel_iffC];
paulson@14387
  1470
paulson@14387
  1471
Goalw [hcomplex_number_of_def] 
paulson@14387
  1472
  "((number_of xa :: hcomplex) + iii * number_of ya = \
paulson@14387
  1473
\       number_of xb) = \
paulson@14387
  1474
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1475
\   ((number_of ya :: hcomplex) = 0))";
paulson@14387
  1476
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2,
paulson@14387
  1477
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
paulson@14387
  1478
qed "hcomplex_number_of_eq_cancel_iff2";
paulson@14387
  1479
Addsimps [hcomplex_number_of_eq_cancel_iff2];
paulson@14387
  1480
paulson@14387
  1481
Goalw [hcomplex_number_of_def] 
paulson@14387
  1482
  "((number_of xa :: hcomplex) + number_of ya * iii = \
paulson@14387
  1483
\       number_of xb) = \
paulson@14387
  1484
\  (((number_of xa :: hcomplex) = number_of xb) & \
paulson@14387
  1485
\   ((number_of ya :: hcomplex) = 0))";
paulson@14387
  1486
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff2a,
paulson@14387
  1487
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
paulson@14387
  1488
qed "hcomplex_number_of_eq_cancel_iff2a";
paulson@14387
  1489
Addsimps [hcomplex_number_of_eq_cancel_iff2a];
paulson@14387
  1490
paulson@14387
  1491
Goalw [hcomplex_number_of_def] 
paulson@14387
  1492
  "((number_of xa :: hcomplex) + iii * number_of ya = \
paulson@14387
  1493
\    iii * number_of yb) = \
paulson@14387
  1494
\  (((number_of xa :: hcomplex) = 0) & \
paulson@14387
  1495
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1496
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3,
paulson@14387
  1497
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
paulson@14387
  1498
qed "hcomplex_number_of_eq_cancel_iff3";
paulson@14387
  1499
Addsimps [hcomplex_number_of_eq_cancel_iff3];
paulson@14387
  1500
paulson@14387
  1501
Goalw [hcomplex_number_of_def] 
paulson@14387
  1502
  "((number_of xa :: hcomplex) + number_of ya * iii= \
paulson@14387
  1503
\    iii * number_of yb) = \
paulson@14387
  1504
\  (((number_of xa :: hcomplex) = 0) & \
paulson@14387
  1505
\   ((number_of ya :: hcomplex) = number_of yb))";
paulson@14387
  1506
by (auto_tac (claset(), HOL_ss addsimps [hcomplex_eq_cancel_iff3a,
paulson@14387
  1507
    hcomplex_hypreal_number_of,hcomplex_of_hypreal_zero_iff]));
paulson@14387
  1508
qed "hcomplex_number_of_eq_cancel_iff3a";
paulson@14387
  1509
Addsimps [hcomplex_number_of_eq_cancel_iff3a];
paulson@14387
  1510
*)
paulson@14387
  1511
paulson@14387
  1512
lemma hcomplex_number_of_hcnj [simp]:
paulson@14387
  1513
     "hcnj (number_of v :: hcomplex) = number_of v"
paulson@14387
  1514
by (simp only: hcomplex_number_of [symmetric] hcomplex_hypreal_number_of
paulson@14387
  1515
               hcomplex_hcnj_hcomplex_of_hypreal)
paulson@14387
  1516
paulson@14387
  1517
lemma hcomplex_number_of_hcmod [simp]: 
paulson@14387
  1518
      "hcmod(number_of v :: hcomplex) = abs (number_of v :: hypreal)"
paulson@14387
  1519
by (simp only: hcomplex_number_of [symmetric] hcomplex_hypreal_number_of
paulson@14387
  1520
               hcmod_hcomplex_of_hypreal)
paulson@14387
  1521
paulson@14387
  1522
lemma hcomplex_number_of_hRe [simp]: 
paulson@14387
  1523
      "hRe(number_of v :: hcomplex) = number_of v"
paulson@14387
  1524
by (simp only: hcomplex_number_of [symmetric] hcomplex_hypreal_number_of
paulson@14387
  1525
               hRe_hcomplex_of_hypreal)
paulson@14387
  1526
paulson@14387
  1527
lemma hcomplex_number_of_hIm [simp]: 
paulson@14387
  1528
      "hIm(number_of v :: hcomplex) = 0"
paulson@14387
  1529
by (simp only: hcomplex_number_of [symmetric] hcomplex_hypreal_number_of
paulson@14387
  1530
               hIm_hcomplex_of_hypreal)
paulson@14387
  1531
paulson@14387
  1532
paulson@14314
  1533
ML
paulson@14314
  1534
{*
paulson@14314
  1535
val hcomplex_zero_def = thm"hcomplex_zero_def";
paulson@14314
  1536
val hcomplex_one_def = thm"hcomplex_one_def";
paulson@14314
  1537
val hcomplex_minus_def = thm"hcomplex_minus_def";
paulson@14314
  1538
val hcomplex_diff_def = thm"hcomplex_diff_def";
paulson@14314
  1539
val hcomplex_divide_def = thm"hcomplex_divide_def";
paulson@14314
  1540
val hcomplex_mult_def = thm"hcomplex_mult_def";
paulson@14314
  1541
val hcomplex_add_def = thm"hcomplex_add_def";
paulson@14314
  1542
val hcomplex_of_complex_def = thm"hcomplex_of_complex_def";
paulson@14314
  1543
val iii_def = thm"iii_def";
paulson@14314
  1544
paulson@14314
  1545
val hcomplexrel_iff = thm"hcomplexrel_iff";
paulson@14314
  1546
val hcomplexrel_refl = thm"hcomplexrel_refl";
paulson@14314
  1547
val hcomplexrel_sym = thm"hcomplexrel_sym";
paulson@14314
  1548
val hcomplexrel_trans = thm"hcomplexrel_trans";
paulson@14314
  1549
val equiv_hcomplexrel = thm"equiv_hcomplexrel";
paulson@14314
  1550
val equiv_hcomplexrel_iff = thm"equiv_hcomplexrel_iff";
paulson@14314
  1551
val hcomplexrel_in_hcomplex = thm"hcomplexrel_in_hcomplex";
paulson@14314
  1552
val inj_on_Abs_hcomplex = thm"inj_on_Abs_hcomplex";
paulson@14314
  1553
val inj_Rep_hcomplex = thm"inj_Rep_hcomplex";
paulson@14314
  1554
val lemma_hcomplexrel_refl = thm"lemma_hcomplexrel_refl";
paulson@14314
  1555
val hcomplex_empty_not_mem = thm"hcomplex_empty_not_mem";
paulson@14314
  1556
val Rep_hcomplex_nonempty = thm"Rep_hcomplex_nonempty";
paulson@14314
  1557
val eq_Abs_hcomplex = thm"eq_Abs_hcomplex";
paulson@14314
  1558
val hRe = thm"hRe";
paulson@14314
  1559
val hIm = thm"hIm";
paulson@14314
  1560
val hcomplex_hRe_hIm_cancel_iff = thm"hcomplex_hRe_hIm_cancel_iff";
paulson@14314
  1561
val hcomplex_hRe_zero = thm"hcomplex_hRe_zero";
paulson@14314
  1562
val hcomplex_hIm_zero = thm"hcomplex_hIm_zero";
paulson@14314
  1563
val hcomplex_hRe_one = thm"hcomplex_hRe_one";
paulson@14314
  1564
val hcomplex_hIm_one = thm"hcomplex_hIm_one";
paulson@14314
  1565
val inj_hcomplex_of_complex = thm"inj_hcomplex_of_complex";
paulson@14314
  1566
val hcomplex_of_complex_i = thm"hcomplex_of_complex_i";
paulson@14314
  1567
val hcomplex_add = thm"hcomplex_add";
paulson@14314
  1568
val hcomplex_add_commute = thm"hcomplex_add_commute";
paulson@14314
  1569
val hcomplex_add_assoc = thm"hcomplex_add_assoc";
paulson@14314
  1570
val hcomplex_add_zero_left = thm"hcomplex_add_zero_left";
paulson@14314
  1571
val hcomplex_add_zero_right = thm"hcomplex_add_zero_right";
paulson@14314
  1572
val hRe_add = thm"hRe_add";
paulson@14314
  1573
val hIm_add = thm"hIm_add";
paulson@14314
  1574
val hcomplex_minus_congruent = thm"hcomplex_minus_congruent";
paulson@14314
  1575
val hcomplex_minus = thm"hcomplex_minus";
paulson@14314
  1576
val hcomplex_add_minus_left = thm"hcomplex_add_minus_left";
paulson@14314
  1577
val hRe_minus = thm"hRe_minus";
paulson@14314
  1578
val hIm_minus = thm"hIm_minus";
paulson@14314
  1579
val hcomplex_add_minus_eq_minus = thm"hcomplex_add_minus_eq_minus";
paulson@14314
  1580
val hcomplex_diff = thm"hcomplex_diff";
paulson@14314
  1581
val hcomplex_diff_eq_eq = thm"hcomplex_diff_eq_eq";
paulson@14314
  1582
val hcomplex_mult = thm"hcomplex_mult";
paulson@14314
  1583
val hcomplex_mult_commute = thm"hcomplex_mult_commute";
paulson@14314
  1584
val hcomplex_mult_assoc = thm"hcomplex_mult_assoc";
paulson@14314
  1585
val hcomplex_mult_one_left = thm"hcomplex_mult_one_left";
paulson@14314
  1586
val hcomplex_mult_one_right = thm"hcomplex_mult_one_right";
paulson@14314
  1587
val hcomplex_mult_zero_left = thm"hcomplex_mult_zero_left";
paulson@14314
  1588
val hcomplex_mult_minus_one = thm"hcomplex_mult_minus_one";
paulson@14314
  1589
val hcomplex_mult_minus_one_right = thm"hcomplex_mult_minus_one_right";
paulson@14314
  1590
val hcomplex_add_mult_distrib = thm"hcomplex_add_mult_distrib";
paulson@14314
  1591
val hcomplex_zero_not_eq_one = thm"hcomplex_zero_not_eq_one";
paulson@14314
  1592
val hcomplex_inverse = thm"hcomplex_inverse";
paulson@14314
  1593
val hcomplex_mult_inv_left = thm"hcomplex_mult_inv_left";
paulson@14314
  1594
val hcomplex_mult_left_cancel = thm"hcomplex_mult_left_cancel";
paulson@14314
  1595
val hcomplex_mult_right_cancel = thm"hcomplex_mult_right_cancel";
paulson@14314
  1596
val hcomplex_add_divide_distrib = thm"hcomplex_add_divide_distrib";
paulson@14314
  1597
val hcomplex_of_hypreal = thm"hcomplex_of_hypreal";
paulson@14314
  1598
val hcomplex_of_hypreal_cancel_iff = thm"hcomplex_of_hypreal_cancel_iff";
paulson@14314
  1599
val hcomplex_of_hypreal_minus = thm"hcomplex_of_hypreal_minus";
paulson@14314
  1600
val hcomplex_of_hypreal_inverse = thm"hcomplex_of_hypreal_inverse";
paulson@14314
  1601
val hcomplex_of_hypreal_add = thm"hcomplex_of_hypreal_add";
paulson@14314
  1602
val hcomplex_of_hypreal_diff = thm"hcomplex_of_hypreal_diff";
paulson@14314
  1603
val hcomplex_of_hypreal_mult = thm"hcomplex_of_hypreal_mult";
paulson@14314
  1604
val hcomplex_of_hypreal_divide = thm"hcomplex_of_hypreal_divide";
paulson@14314
  1605
val hcomplex_of_hypreal_one = thm"hcomplex_of_hypreal_one";
paulson@14314
  1606
val hcomplex_of_hypreal_zero = thm"hcomplex_of_hypreal_zero";
paulson@14314
  1607
val hcomplex_of_hypreal_pow = thm"hcomplex_of_hypreal_pow";
paulson@14314
  1608
val hRe_hcomplex_of_hypreal = thm"hRe_hcomplex_of_hypreal";
paulson@14314
  1609
val hIm_hcomplex_of_hypreal = thm"hIm_hcomplex_of_hypreal";
paulson@14314
  1610
val hcomplex_of_hypreal_epsilon_not_zero = thm"hcomplex_of_hypreal_epsilon_not_zero";
paulson@14314
  1611
val hcmod = thm"hcmod";
paulson@14314
  1612
val hcmod_zero = thm"hcmod_zero";
paulson@14314
  1613
val hcmod_one = thm"hcmod_one";
paulson@14314
  1614
val hcmod_hcomplex_of_hypreal = thm"hcmod_hcomplex_of_hypreal";
paulson@14314
  1615
val hcomplex_of_hypreal_abs = thm"hcomplex_of_hypreal_abs";
paulson@14314
  1616
val hcnj = thm"hcnj";
paulson@14314
  1617
val hcomplex_hcnj_cancel_iff = thm"hcomplex_hcnj_cancel_iff";
paulson@14314
  1618
val hcomplex_hcnj_hcnj = thm"hcomplex_hcnj_hcnj";
paulson@14314
  1619
val hcomplex_hcnj_hcomplex_of_hypreal = thm"hcomplex_hcnj_hcomplex_of_hypreal";
paulson@14314
  1620
val hcomplex_hmod_hcnj = thm"hcomplex_hmod_hcnj";
paulson@14314
  1621
val hcomplex_hcnj_minus = thm"hcomplex_hcnj_minus";
paulson@14314
  1622
val hcomplex_hcnj_inverse = thm"hcomplex_hcnj_inverse";
paulson@14314
  1623
val hcomplex_hcnj_add = thm"hcomplex_hcnj_add";
paulson@14314
  1624
val hcomplex_hcnj_diff = thm"hcomplex_hcnj_diff";
paulson@14314
  1625
val hcomplex_hcnj_mult = thm"hcomplex_hcnj_mult";
paulson@14314
  1626
val hcomplex_hcnj_divide = thm"hcomplex_hcnj_divide";
paulson@14314
  1627
val hcnj_one = thm"hcnj_one";
paulson@14314
  1628
val hcomplex_hcnj_pow = thm"hcomplex_hcnj_pow";
paulson@14314
  1629
val hcomplex_hcnj_zero = thm"hcomplex_hcnj_zero";
paulson@14314
  1630
val hcomplex_hcnj_zero_iff = thm"hcomplex_hcnj_zero_iff";
paulson@14314
  1631
val hcomplex_mult_hcnj = thm"hcomplex_mult_hcnj";
paulson@14314
  1632
val hcomplex_hcmod_eq_zero_cancel = thm"hcomplex_hcmod_eq_zero_cancel";
paulson@14371
  1633
paulson@14314
  1634
val hcmod_hcomplex_of_hypreal_of_nat = thm"hcmod_hcomplex_of_hypreal_of_nat";
paulson@14314
  1635
val hcmod_hcomplex_of_hypreal_of_hypnat = thm"hcmod_hcomplex_of_hypreal_of_hypnat";
paulson@14314
  1636
val hcmod_minus = thm"hcmod_minus";
paulson@14314
  1637
val hcmod_mult_hcnj = thm"hcmod_mult_hcnj";
paulson@14314
  1638
val hcmod_ge_zero = thm"hcmod_ge_zero";
paulson@14314
  1639
val hrabs_hcmod_cancel = thm"hrabs_hcmod_cancel";
paulson@14314
  1640
val hcmod_mult = thm"hcmod_mult";
paulson@14314
  1641
val hcmod_add_squared_eq = thm"hcmod_add_squared_eq";
paulson@14314
  1642
val hcomplex_hRe_mult_hcnj_le_hcmod = thm"hcomplex_hRe_mult_hcnj_le_hcmod";
paulson@14314
  1643
val hcomplex_hRe_mult_hcnj_le_hcmod2 = thm"hcomplex_hRe_mult_hcnj_le_hcmod2";
paulson@14314
  1644
val hcmod_triangle_squared = thm"hcmod_triangle_squared";
paulson@14314
  1645
val hcmod_triangle_ineq = thm"hcmod_triangle_ineq";
paulson@14314
  1646
val hcmod_triangle_ineq2 = thm"hcmod_triangle_ineq2";
paulson@14314
  1647
val hcmod_diff_commute = thm"hcmod_diff_commute";
paulson@14314
  1648
val hcmod_add_less = thm"hcmod_add_less";
paulson@14314
  1649
val hcmod_mult_less = thm"hcmod_mult_less";
paulson@14314
  1650
val hcmod_diff_ineq = thm"hcmod_diff_ineq";
paulson@14314
  1651
val hcpow = thm"hcpow";
paulson@14314
  1652
val hcomplex_of_hypreal_hyperpow = thm"hcomplex_of_hypreal_hyperpow";
paulson@14314
  1653
val hcmod_hcomplexpow = thm"hcmod_hcomplexpow";
paulson@14314
  1654
val hcmod_hcpow = thm"hcmod_hcpow";
paulson@14314
  1655
val hcpow_minus = thm"hcpow_minus";
paulson@14314
  1656
val hcmod_hcomplex_inverse = thm"hcmod_hcomplex_inverse";
paulson@14314
  1657
val hcmod_divide = thm"hcmod_divide";
paulson@14314
  1658
val hcpow_mult = thm"hcpow_mult";
paulson@14314
  1659
val hcpow_zero = thm"hcpow_zero";
paulson@14314
  1660
val hcpow_zero2 = thm"hcpow_zero2";
paulson@14314
  1661
val hcpow_not_zero = thm"hcpow_not_zero";
paulson@14314
  1662
val hcpow_zero_zero = thm"hcpow_zero_zero";
paulson@14314
  1663
val hcomplex_i_mult_eq = thm"hcomplex_i_mult_eq";
paulson@14314
  1664
val hcomplexpow_i_squared = thm"hcomplexpow_i_squared";
paulson@14314
  1665
val hcomplex_i_not_zero = thm"hcomplex_i_not_zero";
paulson@14314
  1666
val hcomplex_divide = thm"hcomplex_divide";
paulson@14314
  1667
val hsgn = thm"hsgn";
paulson@14314
  1668
val hsgn_zero = thm"hsgn_zero";
paulson@14314
  1669
val hsgn_one = thm"hsgn_one";
paulson@14314
  1670
val hsgn_minus = thm"hsgn_minus";
paulson@14314
  1671
val hsgn_eq = thm"hsgn_eq";
paulson@14314
  1672
val lemma_hypreal_P_EX2 = thm"lemma_hypreal_P_EX2";
paulson@14314
  1673
val hcmod_i = thm"hcmod_i";
paulson@14314
  1674
val hcomplex_eq_cancel_iff2 = thm"hcomplex_eq_cancel_iff2";
paulson@14314
  1675
val hRe_hsgn = thm"hRe_hsgn";
paulson@14314
  1676
val hIm_hsgn = thm"hIm_hsgn";
paulson@14314
  1677
val real_two_squares_add_zero_iff = thm"real_two_squares_add_zero_iff";
paulson@14314
  1678
val hRe_mult_i_eq = thm"hRe_mult_i_eq";
paulson@14314
  1679
val hIm_mult_i_eq = thm"hIm_mult_i_eq";
paulson@14314
  1680
val hcmod_mult_i = thm"hcmod_mult_i";
paulson@14314
  1681
val hcmod_mult_i2 = thm"hcmod_mult_i2";
paulson@14314
  1682
val harg = thm"harg";
paulson@14314
  1683
val cos_harg_i_mult_zero = thm"cos_harg_i_mult_zero";
paulson@14314
  1684
val hcomplex_of_hypreal_zero_iff = thm"hcomplex_of_hypreal_zero_iff";
paulson@14314
  1685
val complex_split_polar2 = thm"complex_split_polar2";
paulson@14314
  1686
val hcomplex_split_polar = thm"hcomplex_split_polar";
paulson@14314
  1687
val hcis = thm"hcis";
paulson@14314
  1688
val hcis_eq = thm"hcis_eq";
paulson@14314
  1689
val hrcis = thm"hrcis";
paulson@14314
  1690
val hrcis_Ex = thm"hrcis_Ex";
paulson@14314
  1691
val hRe_hcomplex_polar = thm"hRe_hcomplex_polar";
paulson@14314
  1692
val hRe_hrcis = thm"hRe_hrcis";
paulson@14314
  1693
val hIm_hcomplex_polar = thm"hIm_hcomplex_polar";
paulson@14314
  1694
val hIm_hrcis = thm"hIm_hrcis";
paulson@14314
  1695
val hcmod_complex_polar = thm"hcmod_complex_polar";
paulson@14314
  1696
val hcmod_hrcis = thm"hcmod_hrcis";
paulson@14314
  1697
val hcis_hrcis_eq = thm"hcis_hrcis_eq";
paulson@14314
  1698
val hrcis_mult = thm"hrcis_mult";
paulson@14314
  1699
val hcis_mult = thm"hcis_mult";
paulson@14314
  1700
val hcis_zero = thm"hcis_zero";
paulson@14314
  1701
val hrcis_zero_mod = thm"hrcis_zero_mod";
paulson@14314
  1702
val hrcis_zero_arg = thm"hrcis_zero_arg";
paulson@14314
  1703
val hcomplex_i_mult_minus = thm"hcomplex_i_mult_minus";
paulson@14314
  1704
val hcomplex_i_mult_minus2 = thm"hcomplex_i_mult_minus2";
paulson@14314
  1705
val hcis_hypreal_of_nat_Suc_mult = thm"hcis_hypreal_of_nat_Suc_mult";
paulson@14314
  1706
val NSDeMoivre = thm"NSDeMoivre";
paulson@14314
  1707
val hcis_hypreal_of_hypnat_Suc_mult = thm"hcis_hypreal_of_hypnat_Suc_mult";
paulson@14314
  1708
val NSDeMoivre_ext = thm"NSDeMoivre_ext";
paulson@14314
  1709
val DeMoivre2 = thm"DeMoivre2";
paulson@14314
  1710
val DeMoivre2_ext = thm"DeMoivre2_ext";
paulson@14314
  1711
val hcis_inverse = thm"hcis_inverse";
paulson@14314
  1712
val hrcis_inverse = thm"hrcis_inverse";
paulson@14314
  1713
val hRe_hcis = thm"hRe_hcis";
paulson@14314
  1714
val hIm_hcis = thm"hIm_hcis";
paulson@14314
  1715
val cos_n_hRe_hcis_pow_n = thm"cos_n_hRe_hcis_pow_n";
paulson@14314
  1716
val sin_n_hIm_hcis_pow_n = thm"sin_n_hIm_hcis_pow_n";
paulson@14314
  1717
val cos_n_hRe_hcis_hcpow_n = thm"cos_n_hRe_hcis_hcpow_n";
paulson@14314
  1718
val sin_n_hIm_hcis_hcpow_n = thm"sin_n_hIm_hcis_hcpow_n";
paulson@14314
  1719
val hexpi_add = thm"hexpi_add";
paulson@14314
  1720
val hcomplex_of_complex_add = thm"hcomplex_of_complex_add";
paulson@14314
  1721
val hcomplex_of_complex_mult = thm"hcomplex_of_complex_mult";
paulson@14314
  1722
val hcomplex_of_complex_eq_iff = thm"hcomplex_of_complex_eq_iff";
paulson@14314
  1723
val hcomplex_of_complex_minus = thm"hcomplex_of_complex_minus";
paulson@14314
  1724
val hcomplex_of_complex_one = thm"hcomplex_of_complex_one";
paulson@14314
  1725
val hcomplex_of_complex_zero = thm"hcomplex_of_complex_zero";
paulson@14314
  1726
val hcomplex_of_complex_zero_iff = thm"hcomplex_of_complex_zero_iff";
paulson@14314
  1727
val hcomplex_of_complex_inverse = thm"hcomplex_of_complex_inverse";
paulson@14314
  1728
val hcomplex_of_complex_divide = thm"hcomplex_of_complex_divide";
paulson@14314
  1729
val hRe_hcomplex_of_complex = thm"hRe_hcomplex_of_complex";
paulson@14314
  1730
val hIm_hcomplex_of_complex = thm"hIm_hcomplex_of_complex";
paulson@14314
  1731
val hcmod_hcomplex_of_complex = thm"hcmod_hcomplex_of_complex";
paulson@14314
  1732
*}
paulson@14314
  1733
paulson@13957
  1734
end